/*
 * $Id: PolyUtil.java 5844 2018-05-27 20:32:32Z kredel $
 */

package edu.jas.poly;


import java.util.ArrayList;
import java.util.List;
import java.util.Map;
import java.util.SortedMap;
import java.util.TreeMap;

import org.apache.log4j.Logger;

import edu.jas.arith.BigComplex;
import edu.jas.arith.BigDecimal;
import edu.jas.arith.BigInteger;
import edu.jas.arith.BigRational;
import edu.jas.arith.ModInteger;
import edu.jas.arith.ModIntegerRing;
import edu.jas.arith.Modular;
import edu.jas.arith.ModularRingFactory;
import edu.jas.arith.Product;
import edu.jas.arith.ProductRing;
import edu.jas.arith.Rational;
import edu.jas.structure.Element;
import edu.jas.structure.GcdRingElem;
import edu.jas.structure.RingElem;
import edu.jas.structure.RingFactory;
import edu.jas.structure.UnaryFunctor;
import edu.jas.util.ListUtil;


/**
 * Polynomial utilities, for example conversion between different
 * representations, evaluation and interpolation.
 * @author Heinz Kredel
 */

public class PolyUtil {


    private static final Logger logger = Logger.getLogger(PolyUtil.class);


    private static final boolean debug = logger.isDebugEnabled();


    /**
     * Recursive representation. Represent as polynomial in i variables with
     * coefficients in n-i variables. Works for arbitrary term orders.
     * @param <C> coefficient type.
     * @param rfac recursive polynomial ring factory.
     * @param A polynomial to be converted.
     * @return Recursive represenations of this in the ring rfac.
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursive(
                    GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomial<C> A) {

        GenPolynomial<GenPolynomial<C>> B = rfac.getZERO().copy();
        if (A.isZERO()) {
            return B;
        }
        int i = rfac.nvar;
        GenPolynomial<C> zero = rfac.getZEROCoefficient();
        Map<ExpVector, GenPolynomial<C>> Bv = B.val; //getMap();
        for (Map.Entry<ExpVector, C> y : A.getMap().entrySet()) {
            ExpVector e = y.getKey();
            C a = y.getValue();
            ExpVector f = e.contract(0, i);
            ExpVector g = e.contract(i, e.length() - i);
            GenPolynomial<C> p = Bv.get(f);
            if (p == null) {
                p = zero;
            }
            p = p.sum(a, g);
            Bv.put(f, p);
        }
        return B;
    }


    /**
     * Distribute a recursive polynomial to a generic polynomial. Works for
     * arbitrary term orders.
     * @param <C> coefficient type.
     * @param dfac combined polynomial ring factory of coefficients and this.
     * @param B polynomial to be converted.
     * @return distributed polynomial.
     */
    public static <C extends RingElem<C>> GenPolynomial<C> distribute(GenPolynomialRing<C> dfac,
                    GenPolynomial<GenPolynomial<C>> B) {
        GenPolynomial<C> C = dfac.getZERO().copy();
        if (B.isZERO()) {
            return C;
        }
        Map<ExpVector, C> Cm = C.val; //getMap();
        for (Map.Entry<ExpVector, GenPolynomial<C>> y : B.getMap().entrySet()) {
            ExpVector e = y.getKey();
            GenPolynomial<C> A = y.getValue();
            for (Map.Entry<ExpVector, C> x : A.val.entrySet()) {
                ExpVector f = x.getKey();
                C b = x.getValue();
                ExpVector g = e.combine(f);
                assert (Cm.get(g) == null);
                Cm.put(g, b);
            }
        }
        return C;
    }


    /**
     * Recursive representation. Represent as polynomials in i variables with
     * coefficients in n-i variables. Works for arbitrary term orders.
     * @param <C> coefficient type.
     * @param rfac recursive polynomial ring factory.
     * @param L list of polynomials to be converted.
     * @return Recursive represenations of the list in the ring rfac.
     */
    public static <C extends RingElem<C>> List<GenPolynomial<GenPolynomial<C>>> recursive(
                    GenPolynomialRing<GenPolynomial<C>> rfac, List<GenPolynomial<C>> L) {
        return ListUtil.<GenPolynomial<C>, GenPolynomial<GenPolynomial<C>>> map(L, new DistToRec<C>(rfac));
    }


    /**
     * Distribute a recursive polynomial list to a generic polynomial list.
     * Works for arbitrary term orders.
     * @param <C> coefficient type.
     * @param dfac combined polynomial ring factory of coefficients and this.
     * @param L list of polynomials to be converted.
     * @return distributed polynomial list.
     */
    public static <C extends RingElem<C>> List<GenPolynomial<C>> distribute(GenPolynomialRing<C> dfac,
                    List<GenPolynomial<GenPolynomial<C>>> L) {
        return ListUtil.<GenPolynomial<GenPolynomial<C>>, GenPolynomial<C>> map(L, new RecToDist<C>(dfac));
    }


    /**
     * BigInteger from ModInteger coefficients, symmetric. Represent as
     * polynomial with BigInteger coefficients by removing the modules and
     * making coefficients symmetric to 0.
     * @param fac result polynomial factory.
     * @param A polynomial with ModInteger coefficients to be converted.
     * @return polynomial with BigInteger coefficients.
     */
    public static <C extends RingElem<C> & Modular> GenPolynomial<BigInteger> integerFromModularCoefficients(
                    GenPolynomialRing<BigInteger> fac, GenPolynomial<C> A) {
        return PolyUtil.<C, BigInteger> map(fac, A, new ModSymToInt<C>());
    }


    /**
     * BigInteger from ModInteger coefficients, symmetric. Represent as
     * polynomial with BigInteger coefficients by removing the modules and
     * making coefficients symmetric to 0.
     * @param fac result polynomial factory.
     * @param L list of polynomials with ModInteger coefficients to be
     *            converted.
     * @return list of polynomials with BigInteger coefficients.
     */
    public static <C extends RingElem<C> & Modular> List<GenPolynomial<BigInteger>> integerFromModularCoefficients(
                    final GenPolynomialRing<BigInteger> fac, List<GenPolynomial<C>> L) {
        return ListUtil.<GenPolynomial<C>, GenPolynomial<BigInteger>> map(L,
                        new UnaryFunctor<GenPolynomial<C>, GenPolynomial<BigInteger>>() {


                            public GenPolynomial<BigInteger> eval(GenPolynomial<C> c) {
                                return PolyUtil.<C> integerFromModularCoefficients(fac, c);
                            }
                        });
    }


    /**
     * BigInteger from ModInteger coefficients, positive. Represent as
     * polynomial with BigInteger coefficients by removing the modules.
     * @param fac result polynomial factory.
     * @param A polynomial with ModInteger coefficients to be converted.
     * @return polynomial with BigInteger coefficients.
     */
    public static <C extends RingElem<C> & Modular> GenPolynomial<BigInteger> integerFromModularCoefficientsPositive(
                    GenPolynomialRing<BigInteger> fac, GenPolynomial<C> A) {
        return PolyUtil.<C, BigInteger> map(fac, A, new ModToInt<C>());
    }


    /**
     * BigInteger from BigRational coefficients. Represent as polynomial with
     * BigInteger coefficients by multiplication with the lcm of the numerators
     * of the BigRational coefficients.
     * @param fac result polynomial factory.
     * @param A polynomial with BigRational coefficients to be converted.
     * @return polynomial with BigInteger coefficients.
     */
    public static GenPolynomial<BigInteger> integerFromRationalCoefficients(GenPolynomialRing<BigInteger> fac,
                    GenPolynomial<BigRational> A) {
        if (A == null || A.isZERO()) {
            return fac.getZERO();
        }
        java.math.BigInteger c = null;
        int s = 0;
        // lcm of denominators
        for (BigRational y : A.val.values()) {
            java.math.BigInteger x = y.denominator();
            // c = lcm(c,x)
            if (c == null) {
                c = x;
                s = x.signum();
            } else {
                java.math.BigInteger d = c.gcd(x);
                c = c.multiply(x.divide(d));
            }
        }
        if (s < 0) {
            c = c.negate();
        }
        return PolyUtil.<BigRational, BigInteger> map(fac, A, new RatToInt(c));
    }


    /**
     * BigInteger from BigRational coefficients. Represent as polynomial with
     * BigInteger coefficients by multiplication with the gcd of the numerators
     * and the lcm of the denominators of the BigRational coefficients. <br />
     * <b>Author:</b> Axel Kramer
     * @param fac result polynomial factory.
     * @param A polynomial with BigRational coefficients to be converted.
     * @return Object[] with 3 entries: [0]->gcd [1]->lcm and [2]->polynomial
     *         with BigInteger coefficients.
     */
    public static Object[] integerFromRationalCoefficientsFactor(GenPolynomialRing<BigInteger> fac,
                    GenPolynomial<BigRational> A) {
        Object[] result = new Object[3];
        if (A == null || A.isZERO()) {
            result[0] = java.math.BigInteger.ONE;
            result[1] = java.math.BigInteger.ZERO;
            result[2] = fac.getZERO();
            return result;
        }
        java.math.BigInteger gcd = null;
        java.math.BigInteger lcm = null;
        int sLCM = 0;
        int sGCD = 0;
        // lcm of denominators
        for (BigRational y : A.val.values()) {
            java.math.BigInteger numerator = y.numerator();
            java.math.BigInteger denominator = y.denominator();
            // lcm = lcm(lcm,x)
            if (lcm == null) {
                lcm = denominator;
                sLCM = denominator.signum();
            } else {
                java.math.BigInteger d = lcm.gcd(denominator);
                lcm = lcm.multiply(denominator.divide(d));
            }
            // gcd = gcd(gcd,x)
            if (gcd == null) {
                gcd = numerator;
                sGCD = numerator.signum();
            } else {
                gcd = gcd.gcd(numerator);
            }
        }
        if (sLCM < 0) {
            lcm = lcm.negate();
        }
        if (sGCD < 0) {
            gcd = gcd.negate();
        }
        //System.out.println("gcd* = " + gcd + ", lcm = " + lcm);
        result[0] = gcd;
        result[1] = lcm;
        result[2] = PolyUtil.<BigRational, BigInteger> map(fac, A, new RatToIntFactor(gcd, lcm));
        return result;
    }


    /**
     * BigInteger from BigRational coefficients. Represent as polynomial with
     * BigInteger coefficients by multiplication with the gcd of the numerators
     * and the lcm of the denominators of the BigRational coefficients. <br />
     * @param fac result polynomial factory.
     * @param gcd of rational coefficient numerators.
     * @param lcm of rational coefficient denominators.
     * @param A polynomial with BigRational coefficients to be converted.
     * @return polynomial with BigInteger coefficients.
     */
    public static GenPolynomial<BigInteger> integerFromRationalCoefficients(GenPolynomialRing<BigInteger> fac,
                                                   java.math.BigInteger gcd, java.math.BigInteger lcm,
                                                   GenPolynomial<BigRational> A) {
        //System.out.println("gcd = " + gcd + ", lcm = " + lcm);
        GenPolynomial<BigInteger> Ai = PolyUtil.<BigRational, BigInteger> map(fac, A, new RatToIntFactor(gcd, lcm));
        return Ai;
    }


    /**
     * BigInteger from BigRational coefficients. Represent as list of
     * polynomials with BigInteger coefficients by multiplication with the lcm
     * of the numerators of the BigRational coefficients of each polynomial.
     * @param fac result polynomial factory.
     * @param L list of polynomials with BigRational coefficients to be
     *            converted.
     * @return polynomial list with BigInteger coefficients.
     */
    public static List<GenPolynomial<BigInteger>> integerFromRationalCoefficients(
                    GenPolynomialRing<BigInteger> fac, List<GenPolynomial<BigRational>> L) {
        return ListUtil.<GenPolynomial<BigRational>, GenPolynomial<BigInteger>> map(L, new RatToIntPoly(fac));
    }


    /**
     * From BigInteger coefficients. Represent as polynomial with type C
     * coefficients, e.g. ModInteger or BigRational.
     * @param <C> coefficient type.
     * @param fac result polynomial factory.
     * @param A polynomial with BigInteger coefficients to be converted.
     * @return polynomial with type C coefficients.
     */
    public static <C extends RingElem<C>> GenPolynomial<C> fromIntegerCoefficients(GenPolynomialRing<C> fac,
                    GenPolynomial<BigInteger> A) {
        return PolyUtil.<BigInteger, C> map(fac, A, new FromInteger<C>(fac.coFac));
    }


    /**
     * From BigInteger coefficients. Represent as list of polynomials with type
     * C coefficients, e.g. ModInteger or BigRational.
     * @param <C> coefficient type.
     * @param fac result polynomial factory.
     * @param L list of polynomials with BigInteger coefficients to be
     *            converted.
     * @return list of polynomials with type C coefficients.
     */
    public static <C extends RingElem<C>> List<GenPolynomial<C>> fromIntegerCoefficients(
                    GenPolynomialRing<C> fac, List<GenPolynomial<BigInteger>> L) {
        return ListUtil.<GenPolynomial<BigInteger>, GenPolynomial<C>> map(L, new FromIntegerPoly<C>(fac));
    }


    /**
     * Convert to decimal coefficients.
     * @param fac result polynomial factory.
     * @param A polynomial with Rational coefficients to be converted.
     * @return polynomial with BigDecimal coefficients.
     */
    public static <C extends RingElem<C> & Rational> GenPolynomial<BigDecimal> decimalFromRational(
                    GenPolynomialRing<BigDecimal> fac, GenPolynomial<C> A) {
        return PolyUtil.<C, BigDecimal> map(fac, A, new RatToDec<C>());
    }


    /**
     * Convert to complex decimal coefficients.
     * @param fac result polynomial factory.
     * @param A polynomial with complex Rational coefficients to be converted.
     * @return polynomial with Complex BigDecimal coefficients.
     */
    public static <C extends RingElem<C> & Rational> GenPolynomial<Complex<BigDecimal>> complexDecimalFromRational(
                    GenPolynomialRing<Complex<BigDecimal>> fac, GenPolynomial<Complex<C>> A) {
        return PolyUtil.<Complex<C>, Complex<BigDecimal>> map(fac, A, new CompRatToDec<C>(fac.coFac));
    }


    /**
     * Real part.
     * @param fac result polynomial factory.
     * @param A polynomial with BigComplex coefficients to be converted.
     * @return polynomial with real part of the coefficients.
     */
    public static GenPolynomial<BigRational> realPart(GenPolynomialRing<BigRational> fac,
                    GenPolynomial<BigComplex> A) {
        return PolyUtil.<BigComplex, BigRational> map(fac, A, new RealPart());
    }


    /**
     * Imaginary part.
     * @param fac result polynomial factory.
     * @param A polynomial with BigComplex coefficients to be converted.
     * @return polynomial with imaginary part of coefficients.
     */
    public static GenPolynomial<BigRational> imaginaryPart(GenPolynomialRing<BigRational> fac,
                    GenPolynomial<BigComplex> A) {
        return PolyUtil.<BigComplex, BigRational> map(fac, A, new ImagPart());
    }


    /**
     * Real part.
     * @param fac result polynomial factory.
     * @param A polynomial with BigComplex coefficients to be converted.
     * @return polynomial with real part of the coefficients.
     */
    public static <C extends RingElem<C>> GenPolynomial<C> realPartFromComplex(GenPolynomialRing<C> fac,
                    GenPolynomial<Complex<C>> A) {
        return PolyUtil.<Complex<C>, C> map(fac, A, new RealPartComplex<C>());
    }


    /**
     * Imaginary part.
     * @param fac result polynomial factory.
     * @param A polynomial with BigComplex coefficients to be converted.
     * @return polynomial with imaginary part of coefficients.
     */
    public static <C extends RingElem<C>> GenPolynomial<C> imaginaryPartFromComplex(GenPolynomialRing<C> fac,
                    GenPolynomial<Complex<C>> A) {
        return PolyUtil.<Complex<C>, C> map(fac, A, new ImagPartComplex<C>());
    }


    /**
     * Complex from real polynomial.
     * @param fac result polynomial factory.
     * @param A polynomial with C coefficients to be converted.
     * @return polynomial with Complex<C> coefficients.
     */
    public static <C extends RingElem<C>> GenPolynomial<Complex<C>> toComplex(
                    GenPolynomialRing<Complex<C>> fac, GenPolynomial<C> A) {
        return PolyUtil.<C, Complex<C>> map(fac, A, new ToComplex<C>(fac.coFac));
    }


    /**
     * Complex from rational coefficients.
     * @param fac result polynomial factory.
     * @param A polynomial with BigRational coefficients to be converted.
     * @return polynomial with BigComplex coefficients.
     */
    public static GenPolynomial<BigComplex> complexFromRational(GenPolynomialRing<BigComplex> fac,
                    GenPolynomial<BigRational> A) {
        return PolyUtil.<BigRational, BigComplex> map(fac, A, new RatToCompl());
    }


    /**
     * Complex from ring element coefficients.
     * @param fac result polynomial factory.
     * @param A polynomial with RingElem coefficients to be converted.
     * @return polynomial with Complex coefficients.
     */
    public static <C extends GcdRingElem<C>> GenPolynomial<Complex<C>> complexFromAny(
                    GenPolynomialRing<Complex<C>> fac, GenPolynomial<C> A) {
        ComplexRing<C> cr = (ComplexRing<C>) fac.coFac;
        return PolyUtil.<C, Complex<C>> map(fac, A, new AnyToComplex<C>(cr));
    }


    /**
     * From AlgebraicNumber coefficients. Represent as polynomial with type
     * GenPolynomial&lt;C&gt; coefficients, e.g. ModInteger or BigRational.
     * @param rfac result polynomial factory.
     * @param A polynomial with AlgebraicNumber coefficients to be converted.
     * @return polynomial with type GenPolynomial&lt;C&gt; coefficients.
     */
    public static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>> fromAlgebraicCoefficients(
                    GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomial<AlgebraicNumber<C>> A) {
        return PolyUtil.<AlgebraicNumber<C>, GenPolynomial<C>> map(rfac, A, new AlgToPoly<C>());
    }


    /**
     * Convert to AlgebraicNumber coefficients. Represent as polynomial with
     * AlgebraicNumber<C> coefficients, C is e.g. ModInteger or BigRational.
     * @param pfac result polynomial factory.
     * @param A polynomial with C coefficients to be converted.
     * @return polynomial with AlgebraicNumber&lt;C&gt; coefficients.
     */
    public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> convertToAlgebraicCoefficients(
                    GenPolynomialRing<AlgebraicNumber<C>> pfac, GenPolynomial<C> A) {
        AlgebraicNumberRing<C> afac = (AlgebraicNumberRing<C>) pfac.coFac;
        return PolyUtil.<C, AlgebraicNumber<C>> map(pfac, A, new CoeffToAlg<C>(afac));
    }


    /**
     * Convert to recursive AlgebraicNumber coefficients. Represent as
     * polynomial with recursive AlgebraicNumber<C> coefficients, C is e.g.
     * ModInteger or BigRational.
     * @param depth recursion depth of AlgebraicNumber coefficients.
     * @param pfac result polynomial factory.
     * @param A polynomial with C coefficients to be converted.
     * @return polynomial with AlgebraicNumber&lt;C&gt; coefficients.
     */
    public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> convertToRecAlgebraicCoefficients(
                    int depth, GenPolynomialRing<AlgebraicNumber<C>> pfac, GenPolynomial<C> A) {
        AlgebraicNumberRing<C> afac = (AlgebraicNumberRing<C>) pfac.coFac;
        return PolyUtil.<C, AlgebraicNumber<C>> map(pfac, A, new CoeffToRecAlg<C>(depth, afac));
    }


    /**
     * Convert to AlgebraicNumber coefficients. Represent as polynomial with
     * AlgebraicNumber<C> coefficients, C is e.g. ModInteger or BigRational.
     * @param pfac result polynomial factory.
     * @param A recursive polynomial with GenPolynomial&lt;BigInteger&gt;
     *            coefficients to be converted.
     * @return polynomial with AlgebraicNumber&lt;C&gt; coefficients.
     */
    public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> convertRecursiveToAlgebraicCoefficients(
                    GenPolynomialRing<AlgebraicNumber<C>> pfac, GenPolynomial<GenPolynomial<C>> A) {
        AlgebraicNumberRing<C> afac = (AlgebraicNumberRing<C>) pfac.coFac;
        return PolyUtil.<GenPolynomial<C>, AlgebraicNumber<C>> map(pfac, A, new PolyToAlg<C>(afac));
    }


    /**
     * Complex from algebraic coefficients.
     * @param fac result polynomial factory.
     * @param A polynomial with AlgebraicNumber coefficients Q(i) to be
     *            converted.
     * @return polynomial with Complex coefficients.
     */
    public static <C extends GcdRingElem<C>> GenPolynomial<Complex<C>> complexFromAlgebraic(
                    GenPolynomialRing<Complex<C>> fac, GenPolynomial<AlgebraicNumber<C>> A) {
        ComplexRing<C> cfac = (ComplexRing<C>) fac.coFac;
        return PolyUtil.<AlgebraicNumber<C>, Complex<C>> map(fac, A, new AlgebToCompl<C>(cfac));
    }


    /**
     * AlgebraicNumber from complex coefficients.
     * @param fac result polynomial factory over Q(i).
     * @param A polynomial with Complex coefficients to be converted.
     * @return polynomial with AlgebraicNumber coefficients.
     */
    public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> algebraicFromComplex(
                    GenPolynomialRing<AlgebraicNumber<C>> fac, GenPolynomial<Complex<C>> A) {
        AlgebraicNumberRing<C> afac = (AlgebraicNumberRing<C>) fac.coFac;
        return PolyUtil.<Complex<C>, AlgebraicNumber<C>> map(fac, A, new ComplToAlgeb<C>(afac));
    }


    /**
     * ModInteger chinese remainder algorithm on coefficients.
     * @param fac GenPolynomial&lt;ModInteger&gt; result factory with
     *            A.coFac.modul*B.coFac.modul = C.coFac.modul.
     * @param A GenPolynomial&lt;ModInteger&gt;.
     * @param B other GenPolynomial&lt;ModInteger&gt;.
     * @param mi inverse of A.coFac.modul in ring B.coFac.
     * @return S = cra(A,B), with S mod A.coFac.modul == A and S mod
     *         B.coFac.modul == B.
     */
    public static <C extends RingElem<C> & Modular> GenPolynomial<C> chineseRemainder(
                    GenPolynomialRing<C> fac, GenPolynomial<C> A, C mi, GenPolynomial<C> B) {
        ModularRingFactory<C> cfac = (ModularRingFactory<C>) fac.coFac; // get RingFactory
        GenPolynomial<C> S = fac.getZERO().copy();
        GenPolynomial<C> Ap = A.copy();
        SortedMap<ExpVector, C> av = Ap.val; //getMap();
        SortedMap<ExpVector, C> bv = B.getMap();
        SortedMap<ExpVector, C> sv = S.val; //getMap();
        C c = null;
        for (Map.Entry<ExpVector, C> me : bv.entrySet()) {
            ExpVector e = me.getKey();
            C y = me.getValue(); //bv.get(e); // assert y != null
            C x = av.get(e);
            if (x != null) {
                av.remove(e);
                c = cfac.chineseRemainder(x, mi, y);
                if (!c.isZERO()) { // 0 cannot happen
                    sv.put(e, c);
                }
            } else {
                //c = cfac.fromInteger( y.getVal() );
                c = cfac.chineseRemainder(A.ring.coFac.getZERO(), mi, y);
                if (!c.isZERO()) { // 0 cannot happen
                    sv.put(e, c); // c != null
                }
            }
        }
        // assert bv is empty = done
        for (Map.Entry<ExpVector, C> me : av.entrySet()) { // rest of av
            ExpVector e = me.getKey();
            C x = me.getValue(); // av.get(e); // assert x != null
            //c = cfac.fromInteger( x.getVal() );
            c = cfac.chineseRemainder(x, mi, B.ring.coFac.getZERO());
            if (!c.isZERO()) { // 0 cannot happen
                sv.put(e, c); // c != null
            }
        }
        return S;
    }


    /**
     * GenPolynomial monic, i.e. leadingBaseCoefficient == 1. If
     * leadingBaseCoefficient is not invertible returns this unmodified.
     * @param <C> coefficient type.
     * @param p recursive GenPolynomial<GenPolynomial<C>>.
     * @return monic(p).
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> monic(
                    GenPolynomial<GenPolynomial<C>> p) {
        if (p == null || p.isZERO()) {
            return p;
        }
        C lc = p.leadingBaseCoefficient().leadingBaseCoefficient();
        if (!lc.isUnit()) {
            return p;
        }
        C lm = lc.inverse();
        GenPolynomial<C> L = p.ring.coFac.getONE();
        L = L.multiply(lm);
        return p.multiplyLeft(L);
    }


    /**
     * GenSolvablePolynomial monic, i.e. leadingBaseCoefficient == 1. If
     * leadingBaseCoefficient is not invertible returns this unmodified.
     * @param <C> coefficient type.
     * @param p recursive GenSolvablePolynomial<GenPolynomial<C>>.
     * @return monic(p).
     */
    public static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> monic(
                    GenSolvablePolynomial<GenPolynomial<C>> p) {
        if (p == null || p.isZERO()) {
            return p;
        }
        C lc = p.leadingBaseCoefficient().leadingBaseCoefficient();
        if (!lc.isUnit()) {
            return p;
        }
        C lm = lc.inverse();
        GenPolynomial<C> L = p.ring.coFac.getONE();
        L = L.multiply(lm);
        return p.multiplyLeft(L);
    }


    /**
     * Polynomial list monic.
     * @param <C> coefficient type.
     * @param L list of polynomials with field coefficients.
     * @return list of polynomials with leading coefficient 1.
     */
    public static <C extends RingElem<C>> List<GenPolynomial<C>> monic(List<GenPolynomial<C>> L) {
        return ListUtil.<GenPolynomial<C>, GenPolynomial<C>> map(L,
                        new UnaryFunctor<GenPolynomial<C>, GenPolynomial<C>>() {


                            public GenPolynomial<C> eval(GenPolynomial<C> c) {
                                if (c == null) {
                                    return null;
                                }
                                return c.monic();
                            }
                        });
    }


    /**
     * Word polynomial list monic.
     * @param <C> coefficient type.
     * @param L list of word polynomials with field coefficients.
     * @return list of word polynomials with leading coefficient 1.
     */
    public static <C extends RingElem<C>> List<GenWordPolynomial<C>> wordMonic(List<GenWordPolynomial<C>> L) {
        return ListUtil.<GenWordPolynomial<C>, GenWordPolynomial<C>> map(L,
                        new UnaryFunctor<GenWordPolynomial<C>, GenWordPolynomial<C>>() {


                            public GenWordPolynomial<C> eval(GenWordPolynomial<C> c) {
                                if (c == null) {
                                    return null;
                                }
                                return c.monic();
                            }
                        });
    }


    /**
     * Recursive polynomial list monic.
     * @param <C> coefficient type.
     * @param L list of recursive polynomials with field coefficients.
     * @return list of polynomials with leading base coefficient 1.
     */
    public static <C extends RingElem<C>> List<GenPolynomial<GenPolynomial<C>>> monicRec(
                    List<GenPolynomial<GenPolynomial<C>>> L) {
        return ListUtil.<GenPolynomial<GenPolynomial<C>>, GenPolynomial<GenPolynomial<C>>> map(L,
                        new UnaryFunctor<GenPolynomial<GenPolynomial<C>>, GenPolynomial<GenPolynomial<C>>>() {


                            public GenPolynomial<GenPolynomial<C>> eval(GenPolynomial<GenPolynomial<C>> c) {
                                if (c == null) {
                                    return null;
                                }
                                return PolyUtil.<C> monic(c);
                            }
                        });
    }


    /**
     * Polynomial list leading exponent vectors.
     * @param <C> coefficient type.
     * @param L list of polynomials.
     * @return list of leading exponent vectors.
     */
    public static <C extends RingElem<C>> List<ExpVector> leadingExpVector(List<GenPolynomial<C>> L) {
        return ListUtil.<GenPolynomial<C>, ExpVector> map(L, new UnaryFunctor<GenPolynomial<C>, ExpVector>() {


            public ExpVector eval(GenPolynomial<C> c) {
                if (c == null) {
                    return null;
                }
                return c.leadingExpVector();
            }
        });
    }


    /**
     * Extend coefficient variables. Extend all coefficient ExpVectors by i
     * elements and multiply by x_j^k.
     * @param pfac extended polynomial ring factory (by i variables in the
     *            coefficients).
     * @param j index of variable to be used for multiplication.
     * @param k exponent for x_j.
     * @return extended polynomial.
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> extendCoefficients(
                    GenPolynomialRing<GenPolynomial<C>> pfac, GenPolynomial<GenPolynomial<C>> A, int j,
                    long k) {
        GenPolynomial<GenPolynomial<C>> Cp = pfac.getZERO().copy();
        if (A.isZERO()) {
            return Cp;
        }
        GenPolynomialRing<C> cfac = (GenPolynomialRing<C>) pfac.coFac;
        //GenPolynomialRing<C> acfac = (GenPolynomialRing<C>) A.ring.coFac;
        //int i = cfac.nvar - acfac.nvar;
        Map<ExpVector, GenPolynomial<C>> CC = Cp.val; //getMap();
        for (Map.Entry<ExpVector, GenPolynomial<C>> y : A.val.entrySet()) {
            ExpVector e = y.getKey();
            GenPolynomial<C> a = y.getValue();
            GenPolynomial<C> f = a.extend(cfac, j, k);
            CC.put(e, f);
        }
        return Cp;
    }


    /**
     * Extend coefficient variables. Extend all coefficient ExpVectors by i
     * elements and multiply by x_j^k.
     * @param pfac extended polynomial ring factory (by i variables in the
     *            coefficients).
     * @param j index of variable to be used for multiplication.
     * @param k exponent for x_j.
     * @return extended polynomial.
     */
    public static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> extendCoefficients(
                    GenSolvablePolynomialRing<GenPolynomial<C>> pfac,
                    GenSolvablePolynomial<GenPolynomial<C>> A, int j, long k) {
        GenSolvablePolynomial<GenPolynomial<C>> Cp = pfac.getZERO().copy();
        if (A.isZERO()) {
            return Cp;
        }
        GenPolynomialRing<C> cfac = (GenPolynomialRing<C>) pfac.coFac;
        //GenPolynomialRing<C> acfac = (GenPolynomialRing<C>) A.ring.coFac;
        //int i = cfac.nvar - acfac.nvar;
        Map<ExpVector, GenPolynomial<C>> CC = Cp.val; //getMap();
        for (Map.Entry<ExpVector, GenPolynomial<C>> y : A.val.entrySet()) {
            ExpVector e = y.getKey();
            GenPolynomial<C> a = y.getValue();
            GenPolynomial<C> f = a.extend(cfac, j, k);
            CC.put(e, f);
        }
        return Cp;
    }


    /**
     * To recursive representation. Represent as polynomial in i+r variables
     * with coefficients in i variables. Works for arbitrary term orders.
     * @param <C> coefficient type.
     * @param rfac recursive polynomial ring factory.
     * @param A polynomial to be converted.
     * @return Recursive represenations of A in the ring rfac.
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> toRecursive(
                    GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomial<C> A) {

        GenPolynomial<GenPolynomial<C>> B = rfac.getZERO().copy();
        if (A.isZERO()) {
            return B;
        }
        //int i = rfac.nvar;
        //GenPolynomial<C> zero = rfac.getZEROCoefficient();
        GenPolynomial<C> one = rfac.getONECoefficient();
        Map<ExpVector, GenPolynomial<C>> Bv = B.val; //getMap();
        for (Monomial<C> m : A) {
            ExpVector e = m.e;
            C a = m.c;
            GenPolynomial<C> p = one.multiply(a);
            Bv.put(e, p);
        }
        return B;
    }


    /**
     * To recursive representation. Represent as solvable polynomial in i+r
     * variables with coefficients in i variables. Works for arbitrary term
     * orders.
     * @param <C> coefficient type.
     * @param rfac recursive solvable polynomial ring factory.
     * @param A solvable polynomial to be converted.
     * @return Recursive represenations of A in the ring rfac.
     */
    public static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> toRecursive(
                    GenSolvablePolynomialRing<GenPolynomial<C>> rfac, GenSolvablePolynomial<C> A) {

        GenSolvablePolynomial<GenPolynomial<C>> B = rfac.getZERO().copy();
        if (A.isZERO()) {
            return B;
        }
        //int i = rfac.nvar;
        //GenPolynomial<C> zero = rfac.getZEROCoefficient();
        GenPolynomial<C> one = rfac.getONECoefficient();
        Map<ExpVector, GenPolynomial<C>> Bv = B.val; //getMap();
        for (Monomial<C> m : A) {
            ExpVector e = m.e;
            C a = m.c;
            GenPolynomial<C> p = one.multiply(a);
            Bv.put(e, p);
        }
        return B;
    }


    /**
     * GenPolynomial coefficient wise remainder.
     * @param <C> coefficient type.
     * @param P GenPolynomial.
     * @param s nonzero coefficient.
     * @return coefficient wise remainder.
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> baseRemainderPoly(GenPolynomial<C> P, C s) {
        if (s == null || s.isZERO()) {
            throw new ArithmeticException(P + " division by zero " + s);
        }
        GenPolynomial<C> h = P.ring.getZERO().copy();
        Map<ExpVector, C> hm = h.val; //getMap();
        for (Map.Entry<ExpVector, C> m : P.getMap().entrySet()) {
            ExpVector f = m.getKey();
            C a = m.getValue();
            C x = a.remainder(s);
            hm.put(f, x);
        }
        return h;
    }


    /**
     * GenPolynomial sparse pseudo remainder. For univariate polynomials.
     * @param <C> coefficient type.
     * @param P GenPolynomial.
     * @param S nonzero GenPolynomial.
     * @return remainder with ldcf(S)<sup>m'</sup> P = quotient * S + remainder.
     *         m' &le; deg(P)-deg(S)
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     * @deprecated Use
     *             {@link #baseSparsePseudoRemainder(edu.jas.poly.GenPolynomial,edu.jas.poly.GenPolynomial)}
     *             instead
     */
    @Deprecated
    public static <C extends RingElem<C>> GenPolynomial<C> basePseudoRemainder(GenPolynomial<C> P,
                    GenPolynomial<C> S) {
        return baseSparsePseudoRemainder(P, S);
    }


    /**
     * GenPolynomial sparse pseudo remainder. For univariate polynomials.
     * @param <C> coefficient type.
     * @param P GenPolynomial.
     * @param S nonzero GenPolynomial.
     * @return remainder with ldcf(S)<sup>m'</sup> P = quotient * S + remainder.
     *         m' &le; deg(P)-deg(S)
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> baseSparsePseudoRemainder(GenPolynomial<C> P,
                    GenPolynomial<C> S) {
        if (S == null || S.isZERO()) {
            throw new ArithmeticException(P.toString() + " division by zero " + S);
        }
        if (P.isZERO()) {
            return P;
        }
        if (S.isConstant()) {
            return P.ring.getZERO();
        }
        C c = S.leadingBaseCoefficient();
        ExpVector e = S.leadingExpVector();
        GenPolynomial<C> h;
        GenPolynomial<C> r = P;
        while (!r.isZERO()) {
            ExpVector f = r.leadingExpVector();
            if (f.multipleOf(e)) {
                C a = r.leadingBaseCoefficient();
                f = f.subtract(e);
                C x = a.remainder(c);
                if (x.isZERO()) {
                    C y = a.divide(c);
                    h = S.multiply(y, f); // coeff a
                } else {
                    r = r.multiply(c); // coeff ac
                    h = S.multiply(a, f); // coeff ac
                }
                r = r.subtract(h);
            } else {
                break;
            }
        }
        return r;
    }


    /**
     * GenPolynomial dense pseudo remainder. For univariate polynomials.
     * @param P GenPolynomial.
     * @param S nonzero GenPolynomial.
     * @return remainder with ldcf(S)<sup>m</sup> P = quotient * S + remainder.
     *         m == deg(P)-deg(S)
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> baseDensePseudoRemainder(GenPolynomial<C> P,
                    GenPolynomial<C> S) {
        if (S == null || S.isZERO()) {
            throw new ArithmeticException(P + " division by zero " + S);
        }
        if (P.isZERO()) {
            return P;
        }
        if (S.isConstant()) {
            return P.ring.getZERO();
        }
        long m = P.degree(0);
        long n = S.degree(0);
        C c = S.leadingBaseCoefficient();
        ExpVector e = S.leadingExpVector();
        GenPolynomial<C> h;
        GenPolynomial<C> r = P;
        for (long i = m; i >= n; i--) {
            if (r.isZERO()) {
                return r;
            }
            long k = r.degree(0);
            if (i == k) {
                ExpVector f = r.leadingExpVector();
                C a = r.leadingBaseCoefficient();
                f = f.subtract(e); // EVDIF( f, e );
                //System.out.println("red div = " + f);
                r = r.multiply(c); // coeff ac
                h = S.multiply(a, f); // coeff ac
                r = r.subtract(h);
            } else {
                r = r.multiply(c);
            }
        }
        return r;
    }


    /**
     * GenPolynomial dense pseudo quotient. For univariate polynomials.
     * @param P GenPolynomial.
     * @param S nonzero GenPolynomial.
     * @return quotient with ldcf(S)<sup>m</sup> P = quotient * S + remainder. m
     *         == deg(P)-deg(S)
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> baseDensePseudoQuotient(GenPolynomial<C> P,
                    GenPolynomial<C> S) {
        if (S == null || S.isZERO()) {
            throw new ArithmeticException(P + " division by zero " + S);
        }
        if (P.isZERO()) {
            return P;
        }
        //if (S.degree() <= 0) {
        //    return l^n P; //P.ring.getZERO();
        //}
        long m = P.degree(0);
        long n = S.degree(0);
        C c = S.leadingBaseCoefficient();
        ExpVector e = S.leadingExpVector();
        GenPolynomial<C> q = P.ring.getZERO();
        GenPolynomial<C> h;
        GenPolynomial<C> r = P;
        for (long i = m; i >= n; i--) {
            if (r.isZERO()) {
                return q;
            }
            long k = r.degree(0);
            if (i == k) {
                ExpVector f = r.leadingExpVector();
                C a = r.leadingBaseCoefficient();
                f = f.subtract(e); // EVDIF( f, e );
                //System.out.println("red div = " + f);
                r = r.multiply(c); // coeff ac
                h = S.multiply(a, f); // coeff ac
                r = r.subtract(h);
                q = q.multiply(c);
                q = q.sum(a, f);
            } else {
                q = q.multiply(c);
                r = r.multiply(c);
            }
        }
        return q;
    }


    /**
     * GenPolynomial sparse pseudo divide. For univariate polynomials or exact
     * division.
     * @param <C> coefficient type.
     * @param P GenPolynomial.
     * @param S nonzero GenPolynomial.
     * @return quotient with ldcf(S)<sup>m'</sup> P = quotient * S + remainder.
     *         m' &le; deg(P)-deg(S)
     * @see edu.jas.poly.GenPolynomial#divide(edu.jas.poly.GenPolynomial).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> basePseudoDivide(GenPolynomial<C> P,
                    GenPolynomial<C> S) {
        if (S == null || S.isZERO()) {
            throw new ArithmeticException(P.toString() + " division by zero " + S);
        }
        //if (S.ring.nvar != 1) {
        // ok if exact division
        // throw new RuntimeException(this.getClass().getName()
        //                            + " univariate polynomials only");
        //}
        if (P.isZERO() || S.isONE()) {
            return P;
        }
        C c = S.leadingBaseCoefficient();
        ExpVector e = S.leadingExpVector();
        GenPolynomial<C> h;
        GenPolynomial<C> r = P;
        GenPolynomial<C> q = S.ring.getZERO().copy();

        while (!r.isZERO()) {
            ExpVector f = r.leadingExpVector();
            if (f.multipleOf(e)) {
                C a = r.leadingBaseCoefficient();
                f = f.subtract(e);
                C x = a.remainder(c);
                if (x.isZERO()) {
                    C y = a.divide(c);
                    q = q.sum(y, f);
                    h = S.multiply(y, f); // coeff a
                } else {
                    q = q.multiply(c);
                    q = q.sum(a, f);
                    r = r.multiply(c); // coeff ac
                    h = S.multiply(a, f); // coeff ac
                }
                r = r.subtract(h);
            } else {
                break;
            }
        }
        return q;
    }


    /**
     * GenPolynomial sparse pseudo quotient and remainder. For univariate
     * polynomials or exact division.
     * @param <C> coefficient type.
     * @param P GenPolynomial.
     * @param S nonzero GenPolynomial.
     * @return [ quotient, remainder ] with ldcf(S)<sup>m'</sup> P = quotient *
     *         S + remainder. m' &le; deg(P)-deg(S)
     * @see edu.jas.poly.GenPolynomial#divide(edu.jas.poly.GenPolynomial).
     */
    @SuppressWarnings("unchecked")
    public static <C extends RingElem<C>> GenPolynomial<C>[] basePseudoQuotientRemainder(GenPolynomial<C> P,
                    GenPolynomial<C> S) {
        if (S == null || S.isZERO()) {
            throw new ArithmeticException(P.toString() + " division by zero " + S);
        }
        //if (S.ring.nvar != 1) {
        // ok if exact division
        // throw new RuntimeException(this.getClass().getName()
        //                            + " univariate polynomials only");
        //}
        GenPolynomial<C>[] ret = new GenPolynomial[2];
        ret[0] = null;
        ret[1] = null;
        if (P.isZERO() || S.isONE()) {
            ret[0] = P;
            ret[1] = S.ring.getZERO();
            return ret;
        }
        C c = S.leadingBaseCoefficient();
        ExpVector e = S.leadingExpVector();
        GenPolynomial<C> h;
        GenPolynomial<C> r = P;
        GenPolynomial<C> q = S.ring.getZERO().copy();

        while (!r.isZERO()) {
            ExpVector f = r.leadingExpVector();
            if (f.multipleOf(e)) {
                C a = r.leadingBaseCoefficient();
                f = f.subtract(e);
                C x = a.remainder(c);
                if (x.isZERO()) {
                    C y = a.divide(c);
                    q = q.sum(y, f);
                    h = S.multiply(y, f); // coeff a
                } else {
                    q = q.multiply(c);
                    q = q.sum(a, f);
                    r = r.multiply(c); // coeff a c
                    h = S.multiply(a, f); // coeff c a
                }
                r = r.subtract(h);
            } else {
                break;
            }
        }
        //GenPolynomial<C> rhs = q.multiply(S).sum(r);
        //GenPolynomial<C> lhs = P;
        ret[0] = q;
        ret[1] = r;
        return ret;
    }


    /**
     * Is GenPolynomial pseudo quotient and remainder. For univariate
     * polynomials.
     * @param <C> coefficient type.
     * @param P base GenPolynomial.
     * @param S nonzero base GenPolynomial.
     * @return true, if P = q * S + r, else false.
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     *      <b>Note:</b> not always meaningful and working
     */
    public static <C extends RingElem<C>> boolean isBasePseudoQuotientRemainder(GenPolynomial<C> P,
                    GenPolynomial<C> S, GenPolynomial<C> q, GenPolynomial<C> r) {
        GenPolynomial<C> rhs = q.multiply(S).sum(r);
        //System.out.println("rhs,1 = " + rhs);
        GenPolynomial<C> lhs = P;
        C ldcf = S.leadingBaseCoefficient();
        long d = P.degree(0) - S.degree(0) + 1;
        d = (d > 0 ? d : -d + 2);
        for (long i = 0; i <= d; i++) {
            //System.out.println("lhs-rhs = " + lhs.subtract(rhs));
            if (lhs.equals(rhs) || lhs.negate().equals(rhs)) {
                //System.out.println("lhs,1 = " + lhs);
                return true;
            }
            lhs = lhs.multiply(ldcf);
        }
        GenPolynomial<C> Pp = P;
        rhs = q.multiply(S);
        //System.out.println("rhs,2 = " + rhs);
        for (long i = 0; i <= d; i++) {
            lhs = Pp.subtract(r);
            //System.out.println("lhs-rhs = " + lhs.subtract(rhs));
            if (lhs.equals(rhs) || lhs.negate().equals(rhs)) {
                //System.out.println("lhs,2 = " + lhs);
                return true;
            }
            Pp = Pp.multiply(ldcf);
        }
        C a = P.leadingBaseCoefficient();
        rhs = q.multiply(S).sum(r);
        C b = rhs.leadingBaseCoefficient();
        C gcd = a.gcd(b);
        C p = a.multiply(b);
        C lcm = p.divide(gcd);
        C ap = lcm.divide(a);
        C bp = lcm.divide(b);
        if (P.multiply(ap).equals(rhs.multiply(bp))) {
            return true;
        }
        return false;
    }


    /**
     * GenPolynomial divide. For recursive polynomials. Division by coefficient
     * ring element.
     * @param <C> coefficient type.
     * @param P recursive GenPolynomial.
     * @param s GenPolynomial.
     * @return this/s.
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursiveDivide(
                    GenPolynomial<GenPolynomial<C>> P, GenPolynomial<C> s) {
        if (s == null || s.isZERO()) {
            throw new ArithmeticException("division by zero " + P + ", " + s);
        }
        if (P.isZERO()) {
            return P;
        }
        if (s.isONE()) {
            return P;
        }
        GenPolynomial<GenPolynomial<C>> p = P.ring.getZERO().copy();
        SortedMap<ExpVector, GenPolynomial<C>> pv = p.val; //getMap();
        for (Map.Entry<ExpVector, GenPolynomial<C>> m1 : P.getMap().entrySet()) {
            GenPolynomial<C> c1 = m1.getValue();
            ExpVector e1 = m1.getKey();
            GenPolynomial<C> c = PolyUtil.<C> basePseudoDivide(c1, s);
            if (!c.isZERO()) {
                pv.put(e1, c); // or m1.setValue( c )
            } else {
                System.out.println("rDiv, P  = " + P);
                System.out.println("rDiv, c1 = " + c1);
                System.out.println("rDiv, s  = " + s);
                System.out.println("rDiv, c  = " + c);
                throw new RuntimeException("something is wrong");
            }
        }
        return p;
    }


    /**
     * GenPolynomial divide. For recursive polynomials. Division by coefficient
     * ring element.
     * @param <C> coefficient type.
     * @param P recursive GenPolynomial.
     * @param s GenPolynomial.
     * @return this/s.
     */
    public static <C extends RingElem<C>> GenWordPolynomial<GenPolynomial<C>> recursiveDivide(
                    GenWordPolynomial<GenPolynomial<C>> P, GenPolynomial<C> s) {
        if (s == null || s.isZERO()) {
            throw new ArithmeticException("division by zero " + P + ", " + s);
        }
        if (P.isZERO()) {
            return P;
        }
        if (s.isONE()) {
            return P;
        }
        GenWordPolynomial<GenPolynomial<C>> p = P.ring.getZERO().copy();
        SortedMap<Word, GenPolynomial<C>> pv = p.val; //getMap();
        for (Map.Entry<Word, GenPolynomial<C>> m1 : P.getMap().entrySet()) {
            GenPolynomial<C> c1 = m1.getValue();
            Word e1 = m1.getKey();
            GenPolynomial<C> c = PolyUtil.<C> basePseudoDivide(c1, s);
            if (!c.isZERO()) {
                pv.put(e1, c); // or m1.setValue( c )
            } else {
                System.out.println("rDiv, P  = " + P);
                System.out.println("rDiv, c1 = " + c1);
                System.out.println("rDiv, s  = " + s);
                System.out.println("rDiv, c  = " + c);
                throw new RuntimeException("something is wrong");
            }
        }
        return p;
    }


    /**
     * GenPolynomial base divide. For recursive polynomials. Division by
     * coefficient ring element.
     * @param <C> coefficient type.
     * @param P recursive GenPolynomial.
     * @param s coefficient.
     * @return this/s.
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> baseRecursiveDivide(
                    GenPolynomial<GenPolynomial<C>> P, C s) {
        if (s == null || s.isZERO()) {
            throw new ArithmeticException("division by zero " + P + ", " + s);
        }
        if (P.isZERO()) {
            return P;
        }
        if (s.isONE()) {
            return P;
        }
        GenPolynomial<GenPolynomial<C>> p = P.ring.getZERO().copy();
        SortedMap<ExpVector, GenPolynomial<C>> pv = p.val; //getMap();
        for (Map.Entry<ExpVector, GenPolynomial<C>> m1 : P.getMap().entrySet()) {
            GenPolynomial<C> c1 = m1.getValue();
            ExpVector e1 = m1.getKey();
            GenPolynomial<C> c = PolyUtil.<C> coefficientBasePseudoDivide(c1, s);
            if (!c.isZERO()) {
                pv.put(e1, c); // or m1.setValue( c )
            } else {
                System.out.println("pu, c1 = " + c1);
                System.out.println("pu, s  = " + s);
                System.out.println("pu, c  = " + c);
                throw new RuntimeException("something is wrong");
            }
        }
        return p;
    }


    /**
     * GenPolynomial sparse pseudo remainder. For recursive polynomials.
     * @param <C> coefficient type.
     * @param P recursive GenPolynomial.
     * @param S nonzero recursive GenPolynomial.
     * @return remainder with ldcf(S)<sup>m'</sup> P = quotient * S + remainder.
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     * @deprecated Use
     *             {@link #recursiveSparsePseudoRemainder(edu.jas.poly.GenPolynomial,edu.jas.poly.GenPolynomial)}
     *             instead
     */
    @Deprecated
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursivePseudoRemainder(
                    GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S) {
        return recursiveSparsePseudoRemainder(P, S);
    }


    /**
     * GenPolynomial sparse pseudo remainder. For recursive polynomials.
     * @param <C> coefficient type.
     * @param P recursive GenPolynomial.
     * @param S nonzero recursive GenPolynomial.
     * @return remainder with ldcf(S)<sup>m'</sup> P = quotient * S + remainder.
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursiveSparsePseudoRemainder(
                    GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S) {
        if (S == null || S.isZERO()) {
            throw new ArithmeticException(P + " division by zero " + S);
        }
        if (P == null || P.isZERO()) {
            return P;
        }
        if (S.isConstant()) {
            return P.ring.getZERO();
        }
        GenPolynomial<C> c = S.leadingBaseCoefficient();
        ExpVector e = S.leadingExpVector();
        GenPolynomial<GenPolynomial<C>> h;
        GenPolynomial<GenPolynomial<C>> r = P;
        while (!r.isZERO()) {
            ExpVector f = r.leadingExpVector();
            if (f.multipleOf(e)) {
                GenPolynomial<C> a = r.leadingBaseCoefficient();
                f = f.subtract(e);
                GenPolynomial<C> x = c; //test basePseudoRemainder(a,c);
                if (x.isZERO()) {
                    GenPolynomial<C> y = PolyUtil.<C> basePseudoDivide(a, c);
                    h = S.multiply(y, f); // coeff a
                } else {
                    r = r.multiply(c); // coeff a c
                    h = S.multiply(a, f); // coeff c a
                }
                r = r.subtract(h);
            } else {
                break;
            }
        }
        return r;
    }


    /**
     * GenPolynomial dense pseudo remainder. For recursive polynomials.
     * @param P recursive GenPolynomial.
     * @param S nonzero recursive GenPolynomial.
     * @return remainder with ldcf(S)<sup>m'</sup> P = quotient * S + remainder.
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursiveDensePseudoRemainder(
                    GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S) {
        if (S == null || S.isZERO()) {
            throw new ArithmeticException(P + " division by zero " + S);
        }
        if (P == null || P.isZERO()) {
            return P;
        }
        if (S.isConstant()) {
            return P.ring.getZERO();
        }
        long m = P.degree(0);
        long n = S.degree(0);
        GenPolynomial<C> c = S.leadingBaseCoefficient();
        ExpVector e = S.leadingExpVector();
        GenPolynomial<GenPolynomial<C>> h;
        GenPolynomial<GenPolynomial<C>> r = P;
        for (long i = m; i >= n; i--) {
            if (r.isZERO()) {
                return r;
            }
            long k = r.degree(0);
            if (i == k) {
                ExpVector f = r.leadingExpVector();
                GenPolynomial<C> a = r.leadingBaseCoefficient();
                f = f.subtract(e); //EVDIF( f, e );
                //System.out.println("red div = " + f);
                r = r.multiply(c); // coeff ac
                h = S.multiply(a, f); // coeff ac
                r = r.subtract(h);
            } else {
                r = r.multiply(c);
            }
        }
        return r;
    }


    /**
     * GenPolynomial recursive pseudo divide. For recursive polynomials.
     * @param <C> coefficient type.
     * @param P recursive GenPolynomial.
     * @param S nonzero recursive GenPolynomial.
     * @return quotient with ldcf(S)<sup>m'</sup> P = quotient * S + remainder.
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursivePseudoDivide(
                    GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S) {
        if (S == null || S.isZERO()) {
            throw new ArithmeticException(P + " division by zero " + S);
        }
        //if (S.ring.nvar != 1) {
        // ok if exact division
        // throw new RuntimeException(this.getClass().getName()
        //                            + " univariate polynomials only");
        //}
        if (P == null || P.isZERO()) {
            return P;
        }
        if (S.isONE()) {
            return P;
        }
        GenPolynomial<C> c = S.leadingBaseCoefficient();
        ExpVector e = S.leadingExpVector();
        GenPolynomial<GenPolynomial<C>> h;
        GenPolynomial<GenPolynomial<C>> r = P;
        GenPolynomial<GenPolynomial<C>> q = S.ring.getZERO().copy();
        while (!r.isZERO()) {
            ExpVector f = r.leadingExpVector();
            if (f.multipleOf(e)) {
                GenPolynomial<C> a = r.leadingBaseCoefficient();
                f = f.subtract(e);
                GenPolynomial<C> x = PolyUtil.<C> baseSparsePseudoRemainder(a, c);
                if (x.isZERO() && !c.isConstant()) {
                    GenPolynomial<C> y = PolyUtil.<C> basePseudoDivide(a, c);
                    q = q.sum(y, f);
                    h = S.multiply(y, f); // coeff a
                } else {
                    q = q.multiply(c);
                    q = q.sum(a, f);
                    r = r.multiply(c); // coeff ac
                    h = S.multiply(a, f); // coeff ac
                }
                r = r.subtract(h);
            } else {
                break;
            }
        }
        return q;
    }


    /**
     * Is recursive GenPolynomial pseudo quotient and remainder. For recursive
     * polynomials.
     * @param <C> coefficient type.
     * @param P recursive GenPolynomial.
     * @param S nonzero recursive GenPolynomial.
     * @return true, if P ~= q * S + r, else false.
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     *      <b>Note:</b> not always meaningful and working
     */
    public static <C extends RingElem<C>> boolean isRecursivePseudoQuotientRemainder(
                    GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S,
                    GenPolynomial<GenPolynomial<C>> q, GenPolynomial<GenPolynomial<C>> r) {
        GenPolynomial<GenPolynomial<C>> rhs = q.multiply(S).sum(r);
        GenPolynomial<GenPolynomial<C>> lhs = P;
        GenPolynomial<C> ldcf = S.leadingBaseCoefficient();
        long d = P.degree(0) - S.degree(0) + 1;
        d = (d > 0 ? d : -d + 2);
        for (long i = 0; i <= d; i++) {
            //System.out.println("lhs = " + lhs);
            //System.out.println("rhs = " + rhs);
            //System.out.println("lhs-rhs = " + lhs.subtract(rhs));
            if (lhs.equals(rhs)) {
                return true;
            }
            lhs = lhs.multiply(ldcf);
        }
        GenPolynomial<GenPolynomial<C>> Pp = P;
        rhs = q.multiply(S);
        //System.out.println("rhs,2 = " + rhs);
        for (long i = 0; i <= d; i++) {
            lhs = Pp.subtract(r);
            //System.out.println("lhs-rhs = " + lhs.subtract(rhs));
            if (lhs.equals(rhs)) {
                //System.out.println("lhs,2 = " + lhs);
                return true;
            }
            Pp = Pp.multiply(ldcf);
        }
        GenPolynomial<C> a = P.leadingBaseCoefficient();
        rhs = q.multiply(S).sum(r);
        GenPolynomial<C> b = rhs.leadingBaseCoefficient();
        if (P.multiply(b).equals(rhs.multiply(a))) {
            return true;
        }
        return false;
    }


    /**
     * GenPolynomial pseudo divide. For recursive polynomials.
     * @param <C> coefficient type.
     * @param P recursive GenPolynomial.
     * @param s nonzero GenPolynomial.
     * @return quotient with ldcf(s)<sup>m</sup> P = quotient * s + remainder.
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> coefficientPseudoDivide(
                    GenPolynomial<GenPolynomial<C>> P, GenPolynomial<C> s) {
        if (s == null || s.isZERO()) {
            throw new ArithmeticException(P + " division by zero " + s);
        }
        if (P.isZERO()) {
            return P;
        }
        GenPolynomial<GenPolynomial<C>> p = P.ring.getZERO().copy();
        SortedMap<ExpVector, GenPolynomial<C>> pv = p.val;
        for (Map.Entry<ExpVector, GenPolynomial<C>> m : P.getMap().entrySet()) {
            ExpVector e = m.getKey();
            GenPolynomial<C> c1 = m.getValue();
            GenPolynomial<C> c = basePseudoDivide(c1, s);
            if (debug) {
                GenPolynomial<C> x = c1.remainder(s);
                if (!x.isZERO()) {
                    logger.info("divide x = " + x);
                    throw new ArithmeticException(" no exact division: " + c1 + "/" + s);
                }
            }
            if (c.isZERO()) {
                System.out.println(" no exact division: " + c1 + "/" + s);
                //throw new ArithmeticException(" no exact division: " + c1 + "/" + s);
            } else {
                pv.put(e, c); // or m1.setValue( c )
            }
        }
        return p;
    }


    /**
     * GenPolynomial pseudo divide. For polynomials.
     * @param <C> coefficient type.
     * @param P GenPolynomial.
     * @param s nonzero coefficient.
     * @return quotient with ldcf(s)<sup>m</sup> P = quotient * s + remainder.
     * @see edu.jas.poly.GenPolynomial#remainder(edu.jas.poly.GenPolynomial).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> coefficientBasePseudoDivide(GenPolynomial<C> P,
                    C s) {
        if (s == null || s.isZERO()) {
            throw new ArithmeticException(P + " division by zero " + s);
        }
        if (P.isZERO()) {
            return P;
        }
        GenPolynomial<C> p = P.ring.getZERO().copy();
        SortedMap<ExpVector, C> pv = p.val;
        for (Map.Entry<ExpVector, C> m : P.getMap().entrySet()) {
            ExpVector e = m.getKey();
            C c1 = m.getValue();
            C c = c1.divide(s);
            if (debug) {
                C x = c1.remainder(s);
                if (!x.isZERO()) {
                    logger.info("divide x = " + x);
                    throw new ArithmeticException(" no exact division: " + c1 + "/" + s);
                }
            }
            if (c.isZERO()) {
                System.out.println(" no exact division: " + c1 + "/" + s);
                //throw new ArithmeticException(" no exact division: " + c1 + "/" + s);
            } else {
                pv.put(e, c); // or m1.setValue( c )
            }
        }
        return p;
    }


    /**
     * GenPolynomial polynomial derivative main variable.
     * @param <C> coefficient type.
     * @param P GenPolynomial.
     * @return deriviative(P).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> baseDeriviative(GenPolynomial<C> P) {
        if (P == null || P.isZERO()) {
            return P;
        }
        GenPolynomialRing<C> pfac = P.ring;
        if (pfac.nvar == 0) {
            return pfac.getZERO();
        }
        if (pfac.nvar > 1) {
            // baseContent not possible by return type
            throw new IllegalArgumentException(P.getClass().getName() + " only for univariate polynomials");
        }
        RingFactory<C> rf = pfac.coFac;
        GenPolynomial<C> d = pfac.getZERO().copy();
        Map<ExpVector, C> dm = d.val; //getMap();
        for (Map.Entry<ExpVector, C> m : P.getMap().entrySet()) {
            ExpVector f = m.getKey();
            long fl = f.getVal(0);
            if (fl > 0) {
                C cf = rf.fromInteger(fl);
                C a = m.getValue();
                C x = a.multiply(cf);
                if (x != null && !x.isZERO()) {
                    ExpVector e = ExpVector.create(1, 0, fl - 1L);
                    dm.put(e, x);
                }
            }
        }
        return d;
    }


    /**
     * GenPolynomial polynomial partial derivative variable r.
     * @param <C> coefficient type.
     * @param P GenPolynomial.
     * @param r variable for partial deriviate.
     * @return deriviative(P,r).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> baseDeriviative(GenPolynomial<C> P, int r) {
        if (P == null || P.isZERO()) {
            return P;
        }
        GenPolynomialRing<C> pfac = P.ring;
        if (r < 0 || pfac.nvar <= r) {
            throw new IllegalArgumentException(
                            P.getClass().getName() + " deriviative variable out of bound " + r);
        }
        int rp = pfac.nvar - 1 - r;
        RingFactory<C> rf = pfac.coFac;
        GenPolynomial<C> d = pfac.getZERO().copy();
        Map<ExpVector, C> dm = d.val; //getMap();
        for (Map.Entry<ExpVector, C> m : P.getMap().entrySet()) {
            ExpVector f = m.getKey();
            long fl = f.getVal(rp);
            if (fl > 0) {
                C cf = rf.fromInteger(fl);
                C a = m.getValue();
                C x = a.multiply(cf);
                if (x != null && !x.isZERO()) {
                    ExpVector e = f.subst(rp, fl - 1L);
                    dm.put(e, x);
                }
            }
        }
        return d;
    }


    /**
     * GenPolynomial polynomial integral main variable.
     * @param <C> coefficient type.
     * @param P GenPolynomial.
     * @return integral(P).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> baseIntegral(GenPolynomial<C> P) {
        if (P == null || P.isZERO()) {
            return P;
        }
        GenPolynomialRing<C> pfac = P.ring;
        if (pfac.nvar == 0) {
            return pfac.getONE();
        }
        if (pfac.nvar > 1) {
            // baseContent not possible by return type
            throw new IllegalArgumentException(P.getClass().getName() + " only for univariate polynomials");
        }
        RingFactory<C> rf = pfac.coFac;
        GenPolynomial<C> d = pfac.getZERO().copy();
        Map<ExpVector, C> dm = d.val; //getMap();
        for (Map.Entry<ExpVector, C> m : P.getMap().entrySet()) {
            ExpVector f = m.getKey();
            long fl = f.getVal(0);
            fl++;
            C cf = rf.fromInteger(fl);
            C a = m.getValue();
            C x = a.divide(cf);
            if (x != null && !x.isZERO()) {
                ExpVector e = ExpVector.create(1, 0, fl);
                dm.put(e, x);
            }
        }
        return d;
    }


    /**
     * GenPolynomial recursive polynomial derivative main variable.
     * @param <C> coefficient type.
     * @param P recursive GenPolynomial.
     * @return deriviative(P).
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursiveDeriviative(
                    GenPolynomial<GenPolynomial<C>> P) {
        if (P == null || P.isZERO()) {
            return P;
        }
        GenPolynomialRing<GenPolynomial<C>> pfac = P.ring;
        if (pfac.nvar == 0) {
            return pfac.getZERO();
        }
        if (pfac.nvar > 1) {
            // baseContent not possible by return type
            throw new IllegalArgumentException(P.getClass().getName() + " only for univariate polynomials");
        }
        GenPolynomialRing<C> pr = (GenPolynomialRing<C>) pfac.coFac;
        RingFactory<C> rf = pr.coFac;
        GenPolynomial<GenPolynomial<C>> d = pfac.getZERO().copy();
        Map<ExpVector, GenPolynomial<C>> dm = d.val; //getMap();
        for (Map.Entry<ExpVector, GenPolynomial<C>> m : P.getMap().entrySet()) {
            ExpVector f = m.getKey();
            long fl = f.getVal(0);
            if (fl > 0) {
                C cf = rf.fromInteger(fl);
                GenPolynomial<C> a = m.getValue();
                GenPolynomial<C> x = a.multiply(cf);
                if (x != null && !x.isZERO()) {
                    ExpVector e = ExpVector.create(1, 0, fl - 1L);
                    dm.put(e, x);
                }
            }
        }
        return d;
    }


    /**
     * Factor coefficient bound. See SACIPOL.IPFCB: the product of all maxNorms
     * of potential factors is less than or equal to 2**b times the maxNorm of
     * A.
     * @param e degree vector of a GenPolynomial A.
     * @return 2**b.
     */
    public static BigInteger factorBound(ExpVector e) {
        int n = 0;
        java.math.BigInteger p = java.math.BigInteger.ONE;
        java.math.BigInteger v;
        if (e == null || e.isZERO()) {
            return BigInteger.ONE;
        }
        for (int i = 0; i < e.length(); i++) {
            if (e.getVal(i) > 0) {
                n += (2 * e.getVal(i) - 1);
                v = new java.math.BigInteger("" + (e.getVal(i) - 1));
                p = p.multiply(v);
            }
        }
        n += (p.bitCount() + 1); // log2(p)
        n /= 2;
        v = new java.math.BigInteger("" + 2);
        v = v.shiftLeft(n);
        BigInteger N = new BigInteger(v);
        return N;
    }


    /**
     * Evaluate at main variable.
     * @param <C> coefficient type.
     * @param cfac coefficent polynomial ring factory.
     * @param A recursive polynomial to be evaluated.
     * @param a value to evaluate at.
     * @return A( x_1, ..., x_{n-1}, a ).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> evaluateMainRecursive(GenPolynomialRing<C> cfac,
                    GenPolynomial<GenPolynomial<C>> A, C a) {
        if (A == null || A.isZERO()) {
            return cfac.getZERO();
        }
        if (A.ring.nvar != 1) { 
            throw new IllegalArgumentException("evaluateMain no univariate polynomial");
        }
        if (a == null || a.isZERO()) {
            return A.trailingBaseCoefficient();
        }
        // assert descending exponents, i.e. compatible term order
        Map<ExpVector, GenPolynomial<C>> val = A.getMap();
        GenPolynomial<C> B = null;
        long el1 = -1; // undefined
        long el2 = -1;
        for (Map.Entry<ExpVector, GenPolynomial<C>> me : val.entrySet()) {
            ExpVector e = me.getKey();
            el2 = e.getVal(0);
            if (B == null /*el1 < 0*/) { // first turn
                B = me.getValue(); //val.get(e);
            } else {
                for (long i = el2; i < el1; i++) {
                    B = B.multiply(a);
                }
                B = B.sum(me.getValue()); //val.get(e));
            }
            el1 = el2;
        }
        for (long i = 0; i < el2; i++) {
            B = B.multiply(a);
        }
        return B;
    }


    /**
     * Evaluate at main variable.
     * @param <C> coefficient type.
     * @param cfac coefficent polynomial ring factory.
     * @param A distributed polynomial to be evaluated.
     * @param a value to evaluate at.
     * @return A( x_1, ..., x_{n-1}, a ).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> evaluateMain(GenPolynomialRing<C> cfac,
                    GenPolynomial<C> A, C a) {
        if (A == null || A.isZERO()) {
            return cfac.getZERO();
        }
        GenPolynomialRing<GenPolynomial<C>> rfac = new GenPolynomialRing<GenPolynomial<C>>(cfac, 1);
        if (rfac.nvar + cfac.nvar != A.ring.nvar) {
            throw new IllegalArgumentException("evaluateMain number of variabes mismatch");
        }
        GenPolynomial<GenPolynomial<C>> Ap = recursive(rfac, A);
        return PolyUtil.<C> evaluateMainRecursive(cfac, Ap, a);
    }


    /**
     * Evaluate at main variable.
     * @param <C> coefficient type.
     * @param cfac coefficent ring factory.
     * @param L list of univariate polynomials to be evaluated.
     * @param a value to evaluate at.
     * @return list( A( x_1, ..., x_{n-1}, a ) ) for A in L.
     */
    public static <C extends RingElem<C>> List<GenPolynomial<C>> evaluateMain(GenPolynomialRing<C> cfac,
                    List<GenPolynomial<C>> L, C a) {
        return ListUtil.<GenPolynomial<C>, GenPolynomial<C>> map(L, new EvalMainPol<C>(cfac, a));
    }


    /**
     * Evaluate at main variable.
     * @param <C> coefficient type.
     * @param cfac coefficent ring factory.
     * @param A univariate polynomial to be evaluated.
     * @param a value to evaluate at.
     * @return A( a ).
     */
    public static <C extends RingElem<C>> C evaluateMain(RingFactory<C> cfac, GenPolynomial<C> A, C a) {
        if (A == null || A.isZERO()) {
            return cfac.getZERO();
        }
        if (A.ring.nvar != 1) { 
            throw new IllegalArgumentException("evaluateMain no univariate polynomial");
        }
        if (a == null || a.isZERO()) {
            return A.trailingBaseCoefficient();
        }
        // assert decreasing exponents, i.e. compatible term order
        Map<ExpVector, C> val = A.getMap();
        C B = null;
        long el1 = -1; // undefined
        long el2 = -1;
        for (Map.Entry<ExpVector, C> me : val.entrySet()) {
            ExpVector e = me.getKey();
            el2 = e.getVal(0);
            if (B == null /*el1 < 0*/) { // first turn
                B = me.getValue(); // val.get(e);
            } else {
                for (long i = el2; i < el1; i++) {
                    B = B.multiply(a);
                }
                B = B.sum(me.getValue()); //val.get(e));
            }
            el1 = el2;
        }
        for (long i = 0; i < el2; i++) {
            B = B.multiply(a);
        }
        return B;
    }


    /**
     * Evaluate at main variable.
     * @param <C> coefficient type.
     * @param cfac coefficent ring factory.
     * @param L list of univariate polynomial to be evaluated.
     * @param a value to evaluate at.
     * @return list( A( a ) ) for A in L.
     */
    public static <C extends RingElem<C>> List<C> evaluateMain(RingFactory<C> cfac, List<GenPolynomial<C>> L,
                    C a) {
        return ListUtil.<GenPolynomial<C>, C> map(L, new EvalMain<C>(cfac, a));
    }


    /**
     * Evaluate at k-th variable.
     * @param <C> coefficient type.
     * @param cfac coefficient polynomial ring in k variables C[x_1, ..., x_k]
     *            factory.
     * @param rfac coefficient polynomial ring C[x_1, ..., x_{k-1}] [x_k]
     *            factory, a recursive polynomial ring in 1 variable with
     *            coefficients in k-1 variables.
     * @param nfac polynomial ring in n-1 varaibles C[x_1, ..., x_{k-1}]
     *            [x_{k+1}, ..., x_n] factory, a recursive polynomial ring in
     *            n-k+1 variables with coefficients in k-1 variables.
     * @param dfac polynomial ring in n-1 variables. C[x_1, ..., x_{k-1},
     *            x_{k+1}, ..., x_n] factory.
     * @param A polynomial to be evaluated.
     * @param a value to evaluate at.
     * @return A( x_1, ..., x_{k-1}, a, x_{k+1}, ..., x_n).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> evaluate(GenPolynomialRing<C> cfac,
                    GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomialRing<GenPolynomial<C>> nfac,
                    GenPolynomialRing<C> dfac, GenPolynomial<C> A, C a) {
        if (rfac.nvar != 1) {
            throw new IllegalArgumentException("evaluate coefficient ring not univariate");
        }
        if (A == null || A.isZERO()) {
            return cfac.getZERO();
        }
        Map<ExpVector, GenPolynomial<C>> Ap = A.contract(cfac);
        GenPolynomialRing<C> rcf = (GenPolynomialRing<C>) rfac.coFac;
        GenPolynomial<GenPolynomial<C>> Ev = nfac.getZERO().copy();
        Map<ExpVector, GenPolynomial<C>> Evm = Ev.val; //getMap();
        for (Map.Entry<ExpVector, GenPolynomial<C>> m : Ap.entrySet()) {
            ExpVector e = m.getKey();
            GenPolynomial<C> b = m.getValue();
            GenPolynomial<GenPolynomial<C>> c = recursive(rfac, b);
            GenPolynomial<C> d = evaluateMainRecursive(rcf, c, a);
            if (d != null && !d.isZERO()) {
                Evm.put(e, d);
            }
        }
        GenPolynomial<C> B = distribute(dfac, Ev);
        return B;
    }


    /**
     * Evaluate at first (lowest) variable.
     * @param <C> coefficient type.
     * @param cfac coefficient polynomial ring in first variable C[x_1] factory.
     * @param dfac polynomial ring in n-1 variables. C[x_2, ..., x_n] factory.
     * @param A polynomial to be evaluated.
     * @param a value to evaluate at.
     * @return A( a, x_2, ..., x_n).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> evaluateFirst(GenPolynomialRing<C> cfac,
                    GenPolynomialRing<C> dfac, GenPolynomial<C> A, C a) {
        if (A == null || A.isZERO()) {
            return dfac.getZERO();
        }
        Map<ExpVector, GenPolynomial<C>> Ap = A.contract(cfac);
        //RingFactory<C> rcf = cfac.coFac; // == dfac.coFac

        GenPolynomial<C> B = dfac.getZERO().copy();
        Map<ExpVector, C> Bm = B.val; //getMap();

        for (Map.Entry<ExpVector, GenPolynomial<C>> m : Ap.entrySet()) {
            ExpVector e = m.getKey();
            GenPolynomial<C> b = m.getValue();
            C d = evaluateMain(cfac.coFac, b, a);
            if (d != null && !d.isZERO()) {
                Bm.put(e, d);
            }
        }
        return B;
    }


    /**
     * Evaluate at first (lowest) variable.
     * @param <C> coefficient type. Could also be called evaluateFirst(), but
     *            type erasure of A parameter does not allow same name.
     * @param cfac coefficient polynomial ring in first variable C[x_1] factory.
     * @param dfac polynomial ring in n-1 variables. C[x_2, ..., x_n] factory.
     * @param A recursive polynomial to be evaluated.
     * @param a value to evaluate at.
     * @return A( a, x_2, ..., x_n).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> evaluateFirstRec(GenPolynomialRing<C> cfac,
                    GenPolynomialRing<C> dfac, GenPolynomial<GenPolynomial<C>> A, C a) {
        if (A == null || A.isZERO()) {
            return dfac.getZERO();
        }
        Map<ExpVector, GenPolynomial<C>> Ap = A.getMap();
        GenPolynomial<C> B = dfac.getZERO().copy();
        Map<ExpVector, C> Bm = B.val; //getMap();
        for (Map.Entry<ExpVector, GenPolynomial<C>> m : Ap.entrySet()) {
            ExpVector e = m.getKey();
            GenPolynomial<C> b = m.getValue();
            C d = evaluateMain(cfac.coFac, b, a);
            if (d != null && !d.isZERO()) {
                Bm.put(e, d);
            }
        }
        return B;
    }


    /**
     * Evaluate all variables.
     * @param <C> coefficient type.
     * @param cfac coefficient ring factory.
     * @param A polynomial to be evaluated.
     * @param a = (a_1, a_2, ..., a_n) a tuple of values to evaluate at.
     * @return A(a_1, a_2, ..., a_n).
     */
    public static <C extends RingElem<C>> C evaluateAll(RingFactory<C> cfac, GenPolynomial<C> A, List<C> a) {
        if (A == null || A.isZERO()) {
            return cfac.getZERO();
        }
        GenPolynomialRing<C> dfac = A.ring;
        if (a == null || a.size() != dfac.nvar) {
            throw new IllegalArgumentException("evaluate tuple size not equal to number of variables");
        }
        if (dfac.nvar == 0) {
            return A.trailingBaseCoefficient();
        }
        if (dfac.nvar == 1) {
            return evaluateMain(cfac, A, a.get(0));
        }
        C b = cfac.getZERO();
        GenPolynomial<C> Ap = A;
        for (int k = 0; k < dfac.nvar - 1; k++) {
            C ap = a.get(k);
            GenPolynomialRing<C> c1fac = new GenPolynomialRing<C>(cfac, 1);
            GenPolynomialRing<C> cnfac = new GenPolynomialRing<C>(cfac, dfac.nvar - 1 - k);
            GenPolynomial<C> Bp = evaluateFirst(c1fac, cnfac, Ap, ap);
            if (Bp.isZERO()) {
                return b;
            }
            Ap = Bp;
            //System.out.println("Ap = " + Ap);
        }
        C ap = a.get(dfac.nvar - 1);
        b = evaluateMain(cfac, Ap, ap);
        return b;
    }


    /**
     * Substitute main variable.
     * @param A univariate polynomial.
     * @param s polynomial for substitution.
     * @return polynomial A(x <- s).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> substituteMain(GenPolynomial<C> A,
                    GenPolynomial<C> s) {
        return substituteUnivariate(A, s);
    }


    /**
     * Substitute univariate polynomial.
     * @param f univariate polynomial.
     * @param t polynomial for substitution.
     * @return polynomial f(x <- t).
     */
    public static <C extends RingElem<C>> GenPolynomial<C> substituteUnivariate(GenPolynomial<C> f,
                    GenPolynomial<C> t) {
        if (f == null || t == null) {
            return null;
        }
        GenPolynomialRing<C> fac = f.ring;
        if (fac.nvar > 1) {
            throw new IllegalArgumentException("only for univariate polynomial f");
        }
        if (f.isZERO() || f.isConstant()) {
            return f;
        }
        if (t.ring.nvar > 1) {
            fac = t.ring;
        }
        // assert decending exponents, i.e. compatible term order
        Map<ExpVector, C> val = f.getMap();
        GenPolynomial<C> s = null;
        long el1 = -1; // undefined
        long el2 = -1;
        for (Map.Entry<ExpVector, C> me : val.entrySet()) {
            ExpVector e = me.getKey();
            el2 = e.getVal(0);
            if (s == null /*el1 < 0*/) { // first turn
                s = fac.getZERO().sum(me.getValue()); //val.get(e));
            } else {
                for (long i = el2; i < el1; i++) {
                    s = s.multiply(t);
                }
                s = s.sum(me.getValue()); //val.get(e));
            }
            el1 = el2;
        }
        for (long i = 0; i < el2; i++) {
            s = s.multiply(t);
        }
        //System.out.println("s = " + s);
        return s;
    }


    /**
     * Taylor series for polynomial.
     * @param f univariate polynomial.
     * @param a expansion point.
     * @return Taylor series (a polynomial) of f at a.
     */
    public static <C extends RingElem<C>> GenPolynomial<C> seriesOfTaylor(GenPolynomial<C> f, C a) {
        if (f == null) {
            return null;
        }
        GenPolynomialRing<C> fac = f.ring;
        if (fac.nvar > 1) {
            throw new IllegalArgumentException("only for univariate polynomials");
        }
        if (f.isZERO() || f.isConstant()) {
            return f;
        }
        GenPolynomial<C> s = fac.getZERO();
        C fa = PolyUtil.<C> evaluateMain(fac.coFac, f, a);
        s = s.sum(fa);
        long n = 1;
        long i = 0;
        GenPolynomial<C> g = PolyUtil.<C> baseDeriviative(f);
        //GenPolynomial<C> p = fac.getONE();
        while (!g.isZERO()) {
            i++;
            n *= i;
            fa = PolyUtil.<C> evaluateMain(fac.coFac, g, a);
            GenPolynomial<C> q = fac.univariate(0, i); //p;
            q = q.multiply(fa);
            q = q.divide(fac.fromInteger(n));
            s = s.sum(q);
            g = PolyUtil.<C> baseDeriviative(g);
        }
        //System.out.println("s = " + s);
        return s;
    }


    /**
     * ModInteger interpolate on first variable.
     * @param <C> coefficient type.
     * @param fac GenPolynomial<C> result factory.
     * @param A GenPolynomial.
     * @param M GenPolynomial interpolation modul of A.
     * @param mi inverse of M(am) in ring fac.coFac.
     * @param B evaluation of other GenPolynomial.
     * @param am evaluation point (interpolation modul) of B, i.e. P(am) = B.
     * @return S, with S mod M == A and S(am) == B.
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> interpolate(
                    GenPolynomialRing<GenPolynomial<C>> fac, GenPolynomial<GenPolynomial<C>> A,
                    GenPolynomial<C> M, C mi, GenPolynomial<C> B, C am) {
        GenPolynomial<GenPolynomial<C>> S = fac.getZERO().copy();
        GenPolynomial<GenPolynomial<C>> Ap = A.copy();
        SortedMap<ExpVector, GenPolynomial<C>> av = Ap.val; //getMap();
        SortedMap<ExpVector, C> bv = B.getMap();
        SortedMap<ExpVector, GenPolynomial<C>> sv = S.val; //getMap();
        GenPolynomialRing<C> cfac = (GenPolynomialRing<C>) fac.coFac;
        RingFactory<C> bfac = cfac.coFac;
        GenPolynomial<C> c = null;
        for (Map.Entry<ExpVector, C> me : bv.entrySet()) {
            ExpVector e = me.getKey();
            C y = me.getValue(); //bv.get(e); // assert y != null
            GenPolynomial<C> x = av.get(e);
            if (x != null) {
                av.remove(e);
                c = PolyUtil.<C> interpolate(cfac, x, M, mi, y, am);
                if (!c.isZERO()) { // 0 cannot happen
                    sv.put(e, c);
                }
            } else {
                c = PolyUtil.<C> interpolate(cfac, cfac.getZERO(), M, mi, y, am);
                if (!c.isZERO()) { // 0 cannot happen
                    sv.put(e, c); // c != null
                }
            }
        }
        // assert bv is empty = done
        for (Map.Entry<ExpVector, GenPolynomial<C>> me : av.entrySet()) { // rest of av
            ExpVector e = me.getKey();
            GenPolynomial<C> x = me.getValue(); //av.get(e); // assert x != null
            c = PolyUtil.<C> interpolate(cfac, x, M, mi, bfac.getZERO(), am);
            if (!c.isZERO()) { // 0 cannot happen
                sv.put(e, c); // c != null
            }
        }
        return S;
    }


    /**
     * Univariate polynomial interpolation.
     * @param <C> coefficient type.
     * @param fac GenPolynomial<C> result factory.
     * @param A GenPolynomial.
     * @param M GenPolynomial interpolation modul of A.
     * @param mi inverse of M(am) in ring fac.coFac.
     * @param a evaluation of other GenPolynomial.
     * @param am evaluation point (interpolation modul) of a, i.e. P(am) = a.
     * @return S, with S mod M == A and S(am) == a.
     */
    public static <C extends RingElem<C>> GenPolynomial<C> interpolate(GenPolynomialRing<C> fac,
                    GenPolynomial<C> A, GenPolynomial<C> M, C mi, C a, C am) {
        GenPolynomial<C> s;
        C b = PolyUtil.<C> evaluateMain(fac.coFac, A, am);
        // A mod a.modul
        C d = a.subtract(b); // a-A mod a.modul
        if (d.isZERO()) {
            return A;
        }
        b = d.multiply(mi); // b = (a-A)*mi mod a.modul
        // (M*b)+A mod M = A mod M = 
        // (M*mi*(a-A)+A) mod a.modul = a mod a.modul
        s = M.multiply(b);
        s = s.sum(A);
        return s;
    }


    /**
     * Recursive GenPolynomial switch varaible blocks.
     * @param <C> coefficient type.
     * @param P recursive GenPolynomial in R[X,Y].
     * @return this in R[Y,X].
     */
    public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> switchVariables(
                    GenPolynomial<GenPolynomial<C>> P) {
        if (P == null) {
            throw new IllegalArgumentException("P == null");
        }
        GenPolynomialRing<GenPolynomial<C>> rfac1 = P.ring;
        GenPolynomialRing<C> cfac1 = (GenPolynomialRing<C>) rfac1.coFac;
        GenPolynomialRing<C> cfac2 = new GenPolynomialRing<C>(cfac1.coFac, rfac1);
        GenPolynomial<C> zero = cfac2.getZERO();
        GenPolynomialRing<GenPolynomial<C>> rfac2 = new GenPolynomialRing<GenPolynomial<C>>(cfac2, cfac1);
        GenPolynomial<GenPolynomial<C>> B = rfac2.getZERO().copy();
        if (P.isZERO()) {
            return B;
        }
        for (Monomial<GenPolynomial<C>> mr : P) {
            GenPolynomial<C> cr = mr.c;
            for (Monomial<C> mc : cr) {
                GenPolynomial<C> c = zero.sum(mc.c, mr.e);
                B = B.sum(c, mc.e);
            }
        }
        return B;
    }


    /**
     * Maximal degree of leading terms of a polynomial list.
     * @return maximum degree of the leading terms of a polynomial list.
     */
    public static <C extends RingElem<C>> long totalDegreeLeadingTerm(List<GenPolynomial<C>> P) {
        long degree = 0;
        for (GenPolynomial<C> g : P) {
            long total = g.leadingExpVector().totalDeg();
            if (degree < total) {
                degree = total;
            }
        }
        return degree;
    }


    /**
     * Total degree of polynomial list.
     * @return total degree of the polynomial list.
     */
    public static <C extends RingElem<C>> long totalDegree(List<GenPolynomial<C>> P) {
        long degree = 0;
        for (GenPolynomial<C> g : P) {
            long total = g.totalDegree();
            if (degree < total) {
                degree = total;
            }
        }
        return degree;
    }


    /**
     * Maximal degree of polynomial list.
     * @return maximal degree of the polynomial list.
     */
    public static <C extends RingElem<C>> long maxDegree(List<GenPolynomial<C>> P) {
        long degree = 0;
        for (GenPolynomial<C> g : P) {
            long total = g.degree();
            if (degree < total) {
                degree = total;
            }
        }
        return degree;
    }


    /**
     * Maximal degree in the coefficient polynomials.
     * @param <C> coefficient type.
     * @return maximal degree in the coefficients.
     */
    public static <C extends RingElem<C>> long coeffMaxDegree(GenPolynomial<GenPolynomial<C>> A) {
        if (A.isZERO()) {
            return 0; // 0 or -1 ?;
        }
        long deg = 0;
        for (GenPolynomial<C> a : A.getMap().values()) {
            long d = a.degree();
            if (d > deg) {
                deg = d;
            }
        }
        return deg;
    }


    /**
     * Map a unary function to the coefficients.
     * @param ring result polynomial ring factory.
     * @param p polynomial.
     * @param f evaluation functor.
     * @return new polynomial with coefficients f(p(e)).
     */
    public static <C extends RingElem<C>, D extends RingElem<D>> GenPolynomial<D> map(
                    GenPolynomialRing<D> ring, GenPolynomial<C> p, UnaryFunctor<C, D> f) {
        GenPolynomial<D> n = ring.getZERO().copy();
        SortedMap<ExpVector, D> nv = n.val;
        for (Monomial<C> m : p) {
            D c = f.eval(m.c);
            if (c != null && !c.isZERO()) {
                nv.put(m.e, c);
            }
        }
        return n;
    }


    /**
     * Product representation.
     * @param <C> coefficient type.
     * @param pfac polynomial ring factory.
     * @param L list of polynomials to be represented.
     * @return Product represenation of L in the polynomial ring pfac.
     */
    public static <C extends GcdRingElem<C>> List<GenPolynomial<Product<C>>> toProductGen(
                    GenPolynomialRing<Product<C>> pfac, List<GenPolynomial<C>> L) {

        List<GenPolynomial<Product<C>>> list = new ArrayList<GenPolynomial<Product<C>>>();
        if (L == null || L.size() == 0) {
            return list;
        }
        for (GenPolynomial<C> a : L) {
            GenPolynomial<Product<C>> b = toProductGen(pfac, a);
            list.add(b);
        }
        return list;
    }


    /**
     * Product representation.
     * @param <C> coefficient type.
     * @param pfac polynomial ring factory.
     * @param A polynomial to be represented.
     * @return Product represenation of A in the polynomial ring pfac.
     */
    public static <C extends GcdRingElem<C>> GenPolynomial<Product<C>> toProductGen(
                    GenPolynomialRing<Product<C>> pfac, GenPolynomial<C> A) {

        GenPolynomial<Product<C>> P = pfac.getZERO().copy();
        if (A == null || A.isZERO()) {
            return P;
        }
        RingFactory<Product<C>> rpfac = pfac.coFac;
        ProductRing<C> rfac = (ProductRing<C>) rpfac;
        for (Map.Entry<ExpVector, C> y : A.getMap().entrySet()) {
            ExpVector e = y.getKey();
            C a = y.getValue();
            Product<C> p = toProductGen(rfac, a);
            if (!p.isZERO()) {
                P.doPutToMap(e, p);
            }
        }
        return P;
    }


    /**
     * Product representation.
     * @param <C> coefficient type.
     * @param pfac product ring factory.
     * @param c coefficient to be represented.
     * @return Product represenation of c in the ring pfac.
     */
    public static <C extends GcdRingElem<C>> Product<C> toProductGen(ProductRing<C> pfac, C c) {

        SortedMap<Integer, C> elem = new TreeMap<Integer, C>();
        for (int i = 0; i < pfac.length(); i++) {
            RingFactory<C> rfac = pfac.getFactory(i);
            C u = rfac.copy(c);
            if (u != null && !u.isZERO()) {
                elem.put(i, u);
            }
        }
        return new Product<C>(pfac, elem);
    }


    /**
     * Product representation.
     * @param <C> coefficient type.
     * @param pfac product polynomial ring factory.
     * @param c coefficient to be used.
     * @param e exponent vector.
     * @return Product represenation of c X^e in the ring pfac.
     */
    public static <C extends RingElem<C>> Product<GenPolynomial<C>> toProduct(
                    ProductRing<GenPolynomial<C>> pfac, C c, ExpVector e) {
        SortedMap<Integer, GenPolynomial<C>> elem = new TreeMap<Integer, GenPolynomial<C>>();
        for (int i = 0; i < e.length(); i++) {
            RingFactory<GenPolynomial<C>> rfac = pfac.getFactory(i);
            GenPolynomialRing<C> fac = (GenPolynomialRing<C>) rfac;
            //GenPolynomialRing<C> cfac = fac.ring;
            long a = e.getVal(i);
            GenPolynomial<C> u;
            if (a == 0) {
                u = fac.getONE();
            } else {
                u = fac.univariate(0, a);
            }
            u = u.multiply(c);
            elem.put(i, u);
        }
        return new Product<GenPolynomial<C>>(pfac, elem);
    }


    /**
     * Product representation.
     * @param <C> coefficient type.
     * @param pfac product polynomial ring factory.
     * @param A polynomial.
     * @return Product represenation of the terms of A in the ring pfac.
     */
    public static <C extends RingElem<C>> Product<GenPolynomial<C>> toProduct(
                    ProductRing<GenPolynomial<C>> pfac, GenPolynomial<C> A) {
        Product<GenPolynomial<C>> P = pfac.getZERO();
        if (A == null || A.isZERO()) {
            return P;
        }
        for (Map.Entry<ExpVector, C> y : A.getMap().entrySet()) {
            ExpVector e = y.getKey();
            C a = y.getValue();
            Product<GenPolynomial<C>> p = toProduct(pfac, a, e);
            P = P.sum(p);
        }
        return P;
    }


    /**
     * Product representation.
     * @param pfac product ring factory.
     * @param c coefficient to be represented.
     * @return Product represenation of c in the ring pfac.
     */
    public static Product<ModInteger> toProduct(ProductRing<ModInteger> pfac, BigInteger c) {

        SortedMap<Integer, ModInteger> elem = new TreeMap<Integer, ModInteger>();
        for (int i = 0; i < pfac.length(); i++) {
            RingFactory<ModInteger> rfac = pfac.getFactory(i);
            ModIntegerRing fac = (ModIntegerRing) rfac;
            ModInteger u = fac.fromInteger(c.getVal());
            if (!u.isZERO()) {
                elem.put(i, u);
            }
        }
        return new Product<ModInteger>(pfac, elem);
    }


    /**
     * Product representation.
     * @param pfac polynomial ring factory.
     * @param A polynomial to be represented.
     * @return Product represenation of A in the polynomial ring pfac.
     */
    public static GenPolynomial<Product<ModInteger>> toProduct(GenPolynomialRing<Product<ModInteger>> pfac,
                    GenPolynomial<BigInteger> A) {

        GenPolynomial<Product<ModInteger>> P = pfac.getZERO().copy();
        if (A == null || A.isZERO()) {
            return P;
        }
        RingFactory<Product<ModInteger>> rpfac = pfac.coFac;
        ProductRing<ModInteger> fac = (ProductRing<ModInteger>) rpfac;
        for (Map.Entry<ExpVector, BigInteger> y : A.getMap().entrySet()) {
            ExpVector e = y.getKey();
            BigInteger a = y.getValue();
            Product<ModInteger> p = toProduct(fac, a);
            if (!p.isZERO()) {
                P.doPutToMap(e, p);
            }
        }
        return P;
    }


    /**
     * Product representation.
     * @param pfac polynomial ring factory.
     * @param L list of polynomials to be represented.
     * @return Product represenation of L in the polynomial ring pfac.
     */
    public static List<GenPolynomial<Product<ModInteger>>> toProduct(
                    GenPolynomialRing<Product<ModInteger>> pfac, List<GenPolynomial<BigInteger>> L) {

        List<GenPolynomial<Product<ModInteger>>> list = new ArrayList<GenPolynomial<Product<ModInteger>>>();
        if (L == null || L.size() == 0) {
            return list;
        }
        for (GenPolynomial<BigInteger> a : L) {
            GenPolynomial<Product<ModInteger>> b = toProduct(pfac, a);
            list.add(b);
        }
        return list;
    }


    /**
     * Intersection. Intersection of a list of polynomials with a polynomial
     * ring. The polynomial ring must be a contraction of the polynomial ring of
     * the list of polynomials and the TermOrder must be an elimination order.
     * @param R polynomial ring
     * @param F list of polynomials
     * @return R \cap F
     */
    public static <C extends RingElem<C>> List<GenPolynomial<C>> intersect(GenPolynomialRing<C> R,
                    List<GenPolynomial<C>> F) {
        if (F == null || F.isEmpty()) {
            return F;
        }
        GenPolynomialRing<C> pfac = F.get(0).ring;
        int d = pfac.nvar - R.nvar;
        if (d <= 0) {
            return F;
        }
        List<GenPolynomial<C>> H = new ArrayList<GenPolynomial<C>>(F.size());
        for (GenPolynomial<C> p : F) {
            Map<ExpVector, GenPolynomial<C>> m = null;
            m = p.contract(R);
            if (logger.isDebugEnabled()) {
                logger.debug("intersect contract m = " + m);
            }
            if (m.size() == 1) { // contains one power of variables
                for (Map.Entry<ExpVector, GenPolynomial<C>> me : m.entrySet()) {
                    ExpVector e = me.getKey();
                    if (e.isZERO()) {
                        H.add(me.getValue());
                    }
                }
            }
        }
        GenPolynomialRing<C> tfac = pfac.contract(d);
        if (tfac.equals(R)) { // check 
            return H;
        }
        logger.warn("tfac != R: tfac = " + tfac.toScript() + ", R = " + R.toScript() + ", pfac = "
                        + pfac.toScript());
        // throw new RuntimeException("contract(pfac) != R");
        return H;
    }


    /**
     * Intersection. Intersection of a list of solvable polynomials with a
     * solvable polynomial ring. The solvable polynomial ring must be a
     * contraction of the solvable polynomial ring of the list of polynomials
     * and the TermOrder must be an elimination order.
     * @param R solvable polynomial ring
     * @param F list of solvable polynomials
     * @return R \cap F
     */
    @SuppressWarnings("cast")
    public static <C extends RingElem<C>> List<GenSolvablePolynomial<C>> intersect(
                    GenSolvablePolynomialRing<C> R, List<GenSolvablePolynomial<C>> F) {
        List<GenPolynomial<C>> Fp = PolynomialList.<C> castToList(F);
        GenPolynomialRing<C> Rp = (GenPolynomialRing<C>) R;
        List<GenPolynomial<C>> H = intersect(Rp, Fp);
        return PolynomialList.<C> castToSolvableList(H);
    }


    /**
     * Remove all upper variables which do not occur in polynomial.
     * @param p polynomial.
     * @return polynomial with removed variables
     */
    public static <C extends RingElem<C>> GenPolynomial<C> removeUnusedUpperVariables(GenPolynomial<C> p) {
        GenPolynomialRing<C> fac = p.ring;
        if (fac.nvar <= 1) { // univariate
            return p;
        }
        int[] dep = p.degreeVector().dependencyOnVariables();
        if (fac.nvar == dep.length) { // all variables appear
            return p;
        }
        if (dep.length == 0) { // no variables
            GenPolynomialRing<C> fac0 = new GenPolynomialRing<C>(fac.coFac, 0);
            GenPolynomial<C> p0 = new GenPolynomial<C>(fac0, p.leadingBaseCoefficient());
            return p0;
        }
        int l = dep[0]; // higher variable
        int r = dep[dep.length - 1]; // lower variable
        if (l == 0 /*|| l == fac.nvar-1*/) { // upper variable appears
            return p;
        }
        int n = l;
        GenPolynomialRing<C> facr = fac.contract(n);
        Map<ExpVector, GenPolynomial<C>> mpr = p.contract(facr);
        if (mpr.size() != 1) {
            System.out.println(
                            "upper ex, l = " + l + ", r = " + r + ", p = " + p + ", fac = " + fac.toScript());
            throw new RuntimeException("this should not happen " + mpr);
        }
        GenPolynomial<C> pr = mpr.values().iterator().next();
        n = fac.nvar - 1 - r;
        if (n == 0) {
            return pr;
        } // else case not implemented
        return pr;
    }


    /**
     * Remove all lower variables which do not occur in polynomial.
     * @param p polynomial.
     * @return polynomial with removed variables
     */
    public static <C extends RingElem<C>> GenPolynomial<C> removeUnusedLowerVariables(GenPolynomial<C> p) {
        GenPolynomialRing<C> fac = p.ring;
        if (fac.nvar <= 1) { // univariate
            return p;
        }
        int[] dep = p.degreeVector().dependencyOnVariables();
        if (fac.nvar == dep.length) { // all variables appear
            return p;
        }
        if (dep.length == 0) { // no variables
            GenPolynomialRing<C> fac0 = new GenPolynomialRing<C>(fac.coFac, 0);
            GenPolynomial<C> p0 = new GenPolynomial<C>(fac0, p.leadingBaseCoefficient());
            return p0;
        }
        int l = dep[0]; // higher variable
        int r = dep[dep.length - 1]; // lower variable
        if (r == fac.nvar - 1) { // lower variable appears
            return p;
        }
        int n = r + 1;
        GenPolynomialRing<GenPolynomial<C>> rfac = fac.recursive(n);
        GenPolynomial<GenPolynomial<C>> mpr = recursive(rfac, p);
        if (mpr.length() != p.length()) {
            System.out.println(
                            "lower ex, l = " + l + ", r = " + r + ", p = " + p + ", fac = " + fac.toScript());
            throw new RuntimeException("this should not happen " + mpr);
        }
        RingFactory<C> cf = fac.coFac;
        GenPolynomialRing<C> facl = new GenPolynomialRing<C>(cf, rfac);
        GenPolynomial<C> pr = facl.getZERO().copy();
        for (Monomial<GenPolynomial<C>> m : mpr) {
            ExpVector e = m.e;
            GenPolynomial<C> a = m.c;
            if (!a.isConstant()) {
                throw new RuntimeException("this can not happen " + a);
            }
            C c = a.leadingBaseCoefficient();
            pr.doPutToMap(e, c);
        }
        return pr;
    }


    /**
     * Remove upper block of middle variables which do not occur in polynomial.
     * @param p polynomial.
     * @return polynomial with removed variables
     */
    public static <C extends RingElem<C>> GenPolynomial<C> removeUnusedMiddleVariables(GenPolynomial<C> p) {
        GenPolynomialRing<C> fac = p.ring;
        if (fac.nvar <= 2) { // univariate or bi-variate
            return p;
        }
        int[] dep = p.degreeVector().dependencyOnVariables();
        if (fac.nvar == dep.length) { // all variables appear
            return p;
        }
        if (dep.length == 0) { // no variables
            GenPolynomialRing<C> fac0 = new GenPolynomialRing<C>(fac.coFac, 0);
            GenPolynomial<C> p0 = new GenPolynomial<C>(fac0, p.leadingBaseCoefficient());
            return p0;
        }
        ExpVector e1 = p.leadingExpVector();
        if (dep.length == 1) { // one variable
            TermOrder to = new TermOrder(fac.tord.getEvord());
            int i = dep[0];
            String v1 = e1.indexVarName(i, fac.getVars());
            String[] vars = new String[] { v1 };
            GenPolynomialRing<C> fac1 = new GenPolynomialRing<C>(fac.coFac, to, vars);
            GenPolynomial<C> p1 = fac1.getZERO().copy();
            for (Monomial<C> m : p) {
                ExpVector e = m.e;
                ExpVector f = ExpVector.create(1, 0, e.getVal(i));
                p1.doPutToMap(f, m.c);
            }
            return p1;
        }
        GenPolynomialRing<GenPolynomial<C>> rfac = fac.recursive(1);
        GenPolynomial<GenPolynomial<C>> mpr = recursive(rfac, p);

        int l = dep[0]; // higher variable
        int r = fac.nvar - dep[1]; // next variable
        //System.out.println("l  = " + l);
        //System.out.println("r  = " + r);

        TermOrder to = new TermOrder(fac.tord.getEvord());
        String[] vs = fac.getVars();
        String[] vars = new String[r + 1];
        for (int i = 0; i < r; i++) {
            vars[i] = vs[i];
        }
        vars[r] = e1.indexVarName(l, vs);
        //System.out.println("fac  = " + fac);
        GenPolynomialRing<C> dfac = new GenPolynomialRing<C>(fac.coFac, to, vars);
        //System.out.println("dfac = " + dfac);
        GenPolynomialRing<GenPolynomial<C>> fac2 = dfac.recursive(1);
        GenPolynomialRing<C> cfac = (GenPolynomialRing<C>) fac2.coFac;
        GenPolynomial<GenPolynomial<C>> p2r = fac2.getZERO().copy();
        for (Monomial<GenPolynomial<C>> m : mpr) {
            ExpVector e = m.e;
            GenPolynomial<C> a = m.c;
            Map<ExpVector, GenPolynomial<C>> cc = a.contract(cfac);
            for (Map.Entry<ExpVector, GenPolynomial<C>> me : cc.entrySet()) {
                ExpVector f = me.getKey();
                if (f.isZERO()) {
                    GenPolynomial<C> c = me.getValue(); //cc.get(f);
                    p2r.doPutToMap(e, c);
                } else {
                    throw new RuntimeException("this should not happen " + cc);
                }
            }
        }
        GenPolynomial<C> p2 = distribute(dfac, p2r);
        return p2;
    }


    /**
     * Select polynomial with univariate leading term in variable i.
     * @param i variable index.
     * @return polynomial with head term in variable i
     */
    public static <C extends RingElem<C>> GenPolynomial<C> selectWithVariable(List<GenPolynomial<C>> P,
                    int i) {
        for (GenPolynomial<C> p : P) {
            int[] dep = p.leadingExpVector().dependencyOnVariables();
            if (dep.length == 1 && dep[0] == i) {
                return p;
            }
        }
        return null; // not found       
    }

}


/**
 * Conversion of distributive to recursive representation.
 */
class DistToRec<C extends RingElem<C>>
                implements UnaryFunctor<GenPolynomial<C>, GenPolynomial<GenPolynomial<C>>> {


    GenPolynomialRing<GenPolynomial<C>> fac;


    public DistToRec(GenPolynomialRing<GenPolynomial<C>> fac) {
        this.fac = fac;
    }


    public GenPolynomial<GenPolynomial<C>> eval(GenPolynomial<C> c) {
        if (c == null) {
            return fac.getZERO();
        }
        return PolyUtil.<C> recursive(fac, c);
    }
}


/**
 * Conversion of recursive to distributive representation.
 */
class RecToDist<C extends RingElem<C>>
                implements UnaryFunctor<GenPolynomial<GenPolynomial<C>>, GenPolynomial<C>> {


    GenPolynomialRing<C> fac;


    public RecToDist(GenPolynomialRing<C> fac) {
        this.fac = fac;
    }


    public GenPolynomial<C> eval(GenPolynomial<GenPolynomial<C>> c) {
        if (c == null) {
            return fac.getZERO();
        }
        return PolyUtil.<C> distribute(fac, c);
    }
}


/**
 * BigRational numerator functor.
 */
class RatNumer implements UnaryFunctor<BigRational, BigInteger> {


    public BigInteger eval(BigRational c) {
        if (c == null) {
            return new BigInteger();
        }
        return new BigInteger(c.numerator());
    }
}


/**
 * Conversion of symmetric ModInteger to BigInteger functor.
 */
class ModSymToInt<C extends RingElem<C> & Modular> implements UnaryFunctor<C, BigInteger> {


    public BigInteger eval(C c) {
        if (c == null) {
            return new BigInteger();
        }
        return c.getSymmetricInteger();
    }
}


/**
 * Conversion of ModInteger to BigInteger functor.
 */
class ModToInt<C extends RingElem<C> & Modular> implements UnaryFunctor<C, BigInteger> {


    public BigInteger eval(C c) {
        if (c == null) {
            return new BigInteger();
        }
        return c.getInteger();
    }
}


/**
 * Conversion of BigRational to BigInteger with division by lcm functor. result
 * = num*(lcm/denom).
 */
class RatToInt implements UnaryFunctor<BigRational, BigInteger> {


    java.math.BigInteger lcm;


    public RatToInt(java.math.BigInteger lcm) {
        this.lcm = lcm; //.getVal();
    }


    public BigInteger eval(BigRational c) {
        if (c == null) {
            return new BigInteger();
        }
        // p = num*(lcm/denom)
        java.math.BigInteger b = lcm.divide(c.denominator());
        return new BigInteger(c.numerator().multiply(b));
    }
}


/**
 * Conversion of BigRational to BigInteger. result = (num/gcd)*(lcm/denom).
 */
class RatToIntFactor implements UnaryFunctor<BigRational, BigInteger> {


    final java.math.BigInteger lcm;


    final java.math.BigInteger gcd;


    public RatToIntFactor(java.math.BigInteger gcd, java.math.BigInteger lcm) {
        this.gcd = gcd;
        this.lcm = lcm; // .getVal();
    }


    public BigInteger eval(BigRational c) {
        if (c == null) {
            return new BigInteger();
        }
        if (gcd.equals(java.math.BigInteger.ONE)) {
            // p = num*(lcm/denom)
            java.math.BigInteger b = lcm.divide(c.denominator());
            return new BigInteger(c.numerator().multiply(b));
        }
        // p = (num/gcd)*(lcm/denom)
        java.math.BigInteger a = c.numerator().divide(gcd);
        java.math.BigInteger b = lcm.divide(c.denominator());
        return new BigInteger(a.multiply(b));
    }
}


/**
 * Conversion of Rational to BigDecimal. result = decimal(r).
 */
class RatToDec<C extends Element<C> & Rational> implements UnaryFunctor<C, BigDecimal> {


    public BigDecimal eval(C c) {
        if (c == null) {
            return new BigDecimal();
        }
        return new BigDecimal(c.getRational());
    }
}


/**
 * Conversion of Complex Rational to Complex BigDecimal. result = decimal(r).
 */
class CompRatToDec<C extends RingElem<C> & Rational>
                implements UnaryFunctor<Complex<C>, Complex<BigDecimal>> {


    ComplexRing<BigDecimal> ring;


    public CompRatToDec(RingFactory<Complex<BigDecimal>> ring) {
        this.ring = (ComplexRing<BigDecimal>) ring;
    }


    public Complex<BigDecimal> eval(Complex<C> c) {
        if (c == null) {
            return ring.getZERO();
        }
        BigDecimal r = new BigDecimal(c.getRe().getRational());
        BigDecimal i = new BigDecimal(c.getIm().getRational());
        return new Complex<BigDecimal>(ring, r, i);
    }
}


/**
 * Conversion from BigInteger functor.
 */
class FromInteger<D extends RingElem<D>> implements UnaryFunctor<BigInteger, D> {


    RingFactory<D> ring;


    public FromInteger(RingFactory<D> ring) {
        this.ring = ring;
    }


    public D eval(BigInteger c) {
        if (c == null) {
            return ring.getZERO();
        }
        return ring.fromInteger(c.getVal());
    }
}


/**
 * Conversion from GenPolynomial<BigInteger> functor.
 */
class FromIntegerPoly<D extends RingElem<D>>
                implements UnaryFunctor<GenPolynomial<BigInteger>, GenPolynomial<D>> {


    GenPolynomialRing<D> ring;


    FromInteger<D> fi;


    public FromIntegerPoly(GenPolynomialRing<D> ring) {
        if (ring == null) {
            throw new IllegalArgumentException("ring must not be null");
        }
        this.ring = ring;
        fi = new FromInteger<D>(ring.coFac);
    }


    public GenPolynomial<D> eval(GenPolynomial<BigInteger> c) {
        if (c == null) {
            return ring.getZERO();
        }
        return PolyUtil.<BigInteger, D> map(ring, c, fi);
    }
}


/**
 * Conversion from GenPolynomial<BigRational> to GenPolynomial
 * <BigInteger> functor.
 */
class RatToIntPoly implements UnaryFunctor<GenPolynomial<BigRational>, GenPolynomial<BigInteger>> {


    GenPolynomialRing<BigInteger> ring;


    public RatToIntPoly(GenPolynomialRing<BigInteger> ring) {
        if (ring == null) {
            throw new IllegalArgumentException("ring must not be null");
        }
        this.ring = ring;
    }


    public GenPolynomial<BigInteger> eval(GenPolynomial<BigRational> c) {
        if (c == null) {
            return ring.getZERO();
        }
        return PolyUtil.integerFromRationalCoefficients(ring, c);
    }
}


/**
 * Real part functor.
 */
class RealPart implements UnaryFunctor<BigComplex, BigRational> {


    public BigRational eval(BigComplex c) {
        if (c == null) {
            return new BigRational();
        }
        return c.getRe();
    }
}


/**
 * Imaginary part functor.
 */
class ImagPart implements UnaryFunctor<BigComplex, BigRational> {


    public BigRational eval(BigComplex c) {
        if (c == null) {
            return new BigRational();
        }
        return c.getIm();
    }
}


/**
 * Real part functor.
 */
class RealPartComplex<C extends RingElem<C>> implements UnaryFunctor<Complex<C>, C> {


    public C eval(Complex<C> c) {
        if (c == null) {
            return null;
        }
        return c.getRe();
    }
}


/**
 * Imaginary part functor.
 */
class ImagPartComplex<C extends RingElem<C>> implements UnaryFunctor<Complex<C>, C> {


    public C eval(Complex<C> c) {
        if (c == null) {
            return null;
        }
        return c.getIm();
    }
}


/**
 * Rational to complex functor.
 */
class ToComplex<C extends RingElem<C>> implements UnaryFunctor<C, Complex<C>> {


    final protected ComplexRing<C> cfac;


    @SuppressWarnings("unchecked")
    public ToComplex(RingFactory<Complex<C>> fac) {
        if (fac == null) {
            throw new IllegalArgumentException("fac must not be null");
        }
        cfac = (ComplexRing<C>) fac;
    }


    public Complex<C> eval(C c) {
        if (c == null) {
            return cfac.getZERO();
        }
        return new Complex<C>(cfac, c);
    }
}


/**
 * Rational to complex functor.
 */
class RatToCompl implements UnaryFunctor<BigRational, BigComplex> {


    public BigComplex eval(BigRational c) {
        if (c == null) {
            return new BigComplex();
        }
        return new BigComplex(c);
    }
}


/**
 * Any ring element to generic complex functor.
 */
class AnyToComplex<C extends GcdRingElem<C>> implements UnaryFunctor<C, Complex<C>> {


    final protected ComplexRing<C> cfac;


    public AnyToComplex(ComplexRing<C> fac) {
        if (fac == null) {
            throw new IllegalArgumentException("fac must not be null");
        }
        cfac = fac;
    }


    public AnyToComplex(RingFactory<C> fac) {
        this(new ComplexRing<C>(fac));
    }


    public Complex<C> eval(C a) {
        if (a == null || a.isZERO()) { // should not happen
            return cfac.getZERO();
        } else if (a.isONE()) {
            return cfac.getONE();
        } else {
            return new Complex<C>(cfac, a);
        }
    }
}


/**
 * Algebraic to generic complex functor.
 */
class AlgebToCompl<C extends GcdRingElem<C>> implements UnaryFunctor<AlgebraicNumber<C>, Complex<C>> {


    final protected ComplexRing<C> cfac;


    public AlgebToCompl(ComplexRing<C> fac) {
        if (fac == null) {
            throw new IllegalArgumentException("fac must not be null");
        }
        cfac = fac;
    }


    public Complex<C> eval(AlgebraicNumber<C> a) {
        if (a == null || a.isZERO()) { // should not happen
            return cfac.getZERO();
        } else if (a.isONE()) {
            return cfac.getONE();
        } else {
            GenPolynomial<C> p = a.getVal();
            C real = cfac.ring.getZERO();
            C imag = cfac.ring.getZERO();
            for (Monomial<C> m : p) {
                if (m.exponent().getVal(0) == 1L) {
                    imag = m.coefficient();
                } else if (m.exponent().getVal(0) == 0L) {
                    real = m.coefficient();
                } else {
                    throw new IllegalArgumentException("unexpected monomial " + m);
                }
            }
            //Complex<C> c = new Complex<C>(cfac,real,imag);
            return new Complex<C>(cfac, real, imag);
        }
    }
}


/**
 * Ceneric complex to algebraic number functor.
 */
class ComplToAlgeb<C extends GcdRingElem<C>> implements UnaryFunctor<Complex<C>, AlgebraicNumber<C>> {


    final protected AlgebraicNumberRing<C> afac;


    final protected AlgebraicNumber<C> I;


    public ComplToAlgeb(AlgebraicNumberRing<C> fac) {
        if (fac == null) {
            throw new IllegalArgumentException("fac must not be null");
        }
        afac = fac;
        I = afac.getGenerator();
    }


    public AlgebraicNumber<C> eval(Complex<C> c) {
        if (c == null || c.isZERO()) { // should not happen
            return afac.getZERO();
        } else if (c.isONE()) {
            return afac.getONE();
        } else if (c.isIMAG()) {
            return I;
        } else {
            return I.multiply(c.getIm()).sum(c.getRe());
        }
    }
}


/**
 * Algebraic to polynomial functor.
 */
class AlgToPoly<C extends GcdRingElem<C>> implements UnaryFunctor<AlgebraicNumber<C>, GenPolynomial<C>> {


    public GenPolynomial<C> eval(AlgebraicNumber<C> c) {
        if (c == null) {
            return null;
        }
        return c.val;
    }
}


/**
 * Polynomial to algebriac functor.
 */
class PolyToAlg<C extends GcdRingElem<C>> implements UnaryFunctor<GenPolynomial<C>, AlgebraicNumber<C>> {


    final protected AlgebraicNumberRing<C> afac;


    public PolyToAlg(AlgebraicNumberRing<C> fac) {
        if (fac == null) {
            throw new IllegalArgumentException("fac must not be null");
        }
        afac = fac;
    }


    public AlgebraicNumber<C> eval(GenPolynomial<C> c) {
        if (c == null) {
            return afac.getZERO();
        }
        return new AlgebraicNumber<C>(afac, c);
    }
}


/**
 * Coefficient to algebriac functor.
 */
class CoeffToAlg<C extends GcdRingElem<C>> implements UnaryFunctor<C, AlgebraicNumber<C>> {


    final protected AlgebraicNumberRing<C> afac;


    final protected GenPolynomial<C> zero;


    public CoeffToAlg(AlgebraicNumberRing<C> fac) {
        if (fac == null) {
            throw new IllegalArgumentException("fac must not be null");
        }
        afac = fac;
        GenPolynomialRing<C> pfac = afac.ring;
        zero = pfac.getZERO();
    }


    public AlgebraicNumber<C> eval(C c) {
        if (c == null) {
            return afac.getZERO();
        }
        return new AlgebraicNumber<C>(afac, zero.sum(c));
    }
}


/**
 * Coefficient to recursive algebriac functor.
 */
class CoeffToRecAlg<C extends GcdRingElem<C>> implements UnaryFunctor<C, AlgebraicNumber<C>> {


    final protected List<AlgebraicNumberRing<C>> lfac;


    final int depth;


    @SuppressWarnings({ "unchecked", "cast" })
    public CoeffToRecAlg(int depth, AlgebraicNumberRing<C> fac) {
        if (fac == null) {
            throw new IllegalArgumentException("fac must not be null");
        }
        AlgebraicNumberRing<C> afac = fac;
        this.depth = depth;
        lfac = new ArrayList<AlgebraicNumberRing<C>>(this.depth);
        lfac.add(fac);
        for (int i = 1; i < this.depth; i++) {
            RingFactory<C> rf = afac.ring.coFac;
            if (!(rf instanceof AlgebraicNumberRing)) {
                throw new IllegalArgumentException("fac depth to low");
            }
            afac = (AlgebraicNumberRing<C>) (Object) rf;
            lfac.add(afac);
        }
    }


    @SuppressWarnings({ "unchecked" })
    public AlgebraicNumber<C> eval(C c) {
        if (c == null) {
            return lfac.get(0).getZERO();
        }
        C ac = c;
        AlgebraicNumberRing<C> af = lfac.get(lfac.size() - 1);
        GenPolynomial<C> zero = af.ring.getZERO();
        AlgebraicNumber<C> an = new AlgebraicNumber<C>(af, zero.sum(ac));
        for (int i = lfac.size() - 2; i >= 0; i--) {
            af = lfac.get(i);
            zero = af.ring.getZERO();
            ac = (C) (Object) an;
            an = new AlgebraicNumber<C>(af, zero.sum(ac));
        }
        return an;
    }
}


/**
 * Evaluate main variable functor.
 */
class EvalMain<C extends RingElem<C>> implements UnaryFunctor<GenPolynomial<C>, C> {


    final RingFactory<C> cfac;


    final C a;


    public EvalMain(RingFactory<C> cfac, C a) {
        this.cfac = cfac;
        this.a = a;
    }


    public C eval(GenPolynomial<C> c) {
        if (c == null) {
            return cfac.getZERO();
        }
        return PolyUtil.<C> evaluateMain(cfac, c, a);
    }
}


/**
 * Evaluate main variable functor.
 */
class EvalMainPol<C extends RingElem<C>> implements UnaryFunctor<GenPolynomial<C>, GenPolynomial<C>> {


    final GenPolynomialRing<C> cfac;


    final C a;


    public EvalMainPol(GenPolynomialRing<C> cfac, C a) {
        this.cfac = cfac;
        this.a = a;
    }


    public GenPolynomial<C> eval(GenPolynomial<C> c) {
        if (c == null) {
            return cfac.getZERO();
        }
        return PolyUtil.<C> evaluateMain(cfac, c, a);
    }
}