PolyUtil

java.lang.Object
- edu.jas.poly.PolyUtil

```
public class PolyUtil
extends java.lang.Object
```
Polynomial utilities, for example conversion between different representations, evaluation and interpolation.

Author:

Heinz Kredel

Constructor Summary

Constructors
Constructor and Description

PolyUtil()

Constructors
Constructor and Description
`PolyUtil()`

Method Summary

All Methods Static Methods Concrete Methods Deprecated Methods
Modifier and Type	Method and Description
`static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>>`	`algebraicFromComplex(GenPolynomialRing<AlgebraicNumber<C>> fac, GenPolynomial<Complex<C>> A)` AlgebraicNumber from complex coefficients.
`static <C extends RingElem<C>> GenPolynomial<C>`	`baseDensePseudoQuotient(GenPolynomial<C> P, GenPolynomial<C> S)` GenPolynomial dense pseudo quotient.
`static <C extends RingElem<C>> GenPolynomial<C>`	`baseDensePseudoRemainder(GenPolynomial<C> P, GenPolynomial<C> S)` GenPolynomial dense pseudo remainder.
`static <C extends RingElem<C>> GenPolynomial<C>`	`baseDeriviative(GenPolynomial<C> P)` GenPolynomial polynomial derivative main variable.
`static <C extends RingElem<C>> GenPolynomial<C>`	`baseDeriviative(GenPolynomial<C> P, int r)` GenPolynomial polynomial partial derivative variable r.
`static <C extends RingElem<C>> GenPolynomial<C>`	`baseIntegral(GenPolynomial<C> P)` GenPolynomial polynomial integral main variable.
`static <C extends RingElem<C>> GenPolynomial<C>`	`basePseudoDivide(GenPolynomial<C> P, GenPolynomial<C> S)` GenPolynomial sparse pseudo divide.
`static <C extends RingElem<C>> GenPolynomial<C>[]`	`basePseudoQuotientRemainder(GenPolynomial<C> P, GenPolynomial<C> S)` GenPolynomial sparse pseudo quotient and remainder.
`static <C extends RingElem<C>> GenPolynomial<C>`	`basePseudoRemainder(GenPolynomial<C> P, GenPolynomial<C> S)` Deprecated. Use `baseSparsePseudoRemainder(edu.jas.poly.GenPolynomial,edu.jas.poly.GenPolynomial)` instead
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`baseRecursiveDivide(GenPolynomial<GenPolynomial<C>> P, C s)` GenPolynomial base divide.
`static <C extends RingElem<C>> GenPolynomial<C>`	`baseRemainderPoly(GenPolynomial<C> P, C s)` GenPolynomial coefficient wise remainder.
`static <C extends RingElem<C>> GenPolynomial<C>`	`baseSparsePseudoRemainder(GenPolynomial<C> P, GenPolynomial<C> S)` GenPolynomial sparse pseudo remainder.
`static <C extends RingElem<C> & Modular> GenPolynomial<C>`	`chineseRemainder(GenPolynomialRing<C> fac, GenPolynomial<C> A, C mi, GenPolynomial<C> B)` ModInteger chinese remainder algorithm on coefficients.
`static <C extends RingElem<C>> GenPolynomial<C>`	`coefficientBasePseudoDivide(GenPolynomial<C> P, C s)` GenPolynomial pseudo divide.
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`coefficientPseudoDivide(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<C> s)` GenPolynomial pseudo divide.
`static <C extends RingElem<C>> long`	`coeffMaxDegree(GenPolynomial<GenPolynomial<C>> A)` Maximal degree in the coefficient polynomials.
`static <C extends RingElem<C> & Rational> GenPolynomial<Complex<BigDecimal>>`	`complexDecimalFromRational(GenPolynomialRing<Complex<BigDecimal>> fac, GenPolynomial<Complex<C>> A)` Convert to complex decimal coefficients.
`static <C extends GcdRingElem<C>> GenPolynomial<Complex<C>>`	`complexFromAlgebraic(GenPolynomialRing<Complex<C>> fac, GenPolynomial<AlgebraicNumber<C>> A)` Complex from algebraic coefficients.
`static <C extends GcdRingElem<C>> GenPolynomial<Complex<C>>`	`complexFromAny(GenPolynomialRing<Complex<C>> fac, GenPolynomial<C> A)` Complex from ring element coefficients.
`static GenPolynomial<BigComplex>`	`complexFromRational(GenPolynomialRing<BigComplex> fac, GenPolynomial<BigRational> A)` Complex from rational coefficients.
`static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>>`	`convertRecursiveToAlgebraicCoefficients(GenPolynomialRing<AlgebraicNumber<C>> pfac, GenPolynomial<GenPolynomial<C>> A)` Convert to AlgebraicNumber coefficients.
`static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>>`	`convertToAlgebraicCoefficients(GenPolynomialRing<AlgebraicNumber<C>> pfac, GenPolynomial<C> A)` Convert to AlgebraicNumber coefficients.
`static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>>`	`convertToRecAlgebraicCoefficients(int depth, GenPolynomialRing<AlgebraicNumber<C>> pfac, GenPolynomial<C> A)` Convert to recursive AlgebraicNumber coefficients.
`static <C extends RingElem<C> & Rational> GenPolynomial<BigDecimal>`	`decimalFromRational(GenPolynomialRing<BigDecimal> fac, GenPolynomial<C> A)` Convert to decimal coefficients.
`static <C extends RingElem<C>> GenPolynomial<C>`	`distribute(GenPolynomialRing<C> dfac, GenPolynomial<GenPolynomial<C>> B)` Distribute a recursive polynomial to a generic polynomial.
`static <C extends RingElem<C>> java.util.List<GenPolynomial<C>>`	`distribute(GenPolynomialRing<C> dfac, java.util.List<GenPolynomial<GenPolynomial<C>>> L)` Distribute a recursive polynomial list to a generic polynomial list.
`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)` Evaluate at k-th variable.
`static <C extends RingElem<C>> C`	`evaluateAll(RingFactory<C> cfac, GenPolynomial<C> A, java.util.List<C> a)` Evaluate all variables.
`static <C extends RingElem<C>> C`	`evaluateAll(RingFactory<C> cfac, GenPolynomialRing<C> dfac, GenPolynomial<C> A, java.util.List<C> a)` Deprecated. use evaluateAll() with three arguments
`static <C extends RingElem<C>> GenPolynomial<C>`	`evaluateFirst(GenPolynomialRing<C> cfac, GenPolynomialRing<C> dfac, GenPolynomial<C> A, C a)` Evaluate at first (lowest) variable.
`static <C extends RingElem<C>> GenPolynomial<C>`	`evaluateFirstRec(GenPolynomialRing<C> cfac, GenPolynomialRing<C> dfac, GenPolynomial<GenPolynomial<C>> A, C a)` Evaluate at first (lowest) variable.
`static <C extends RingElem<C>> GenPolynomial<C>`	`evaluateMain(GenPolynomialRing<C> cfac, GenPolynomial<C> A, C a)` Evaluate at main variable.
`static <C extends RingElem<C>> java.util.List<GenPolynomial<C>>`	`evaluateMain(GenPolynomialRing<C> cfac, java.util.List<GenPolynomial<C>> L, C a)` Evaluate at main variable.
`static <C extends RingElem<C>> C`	`evaluateMain(RingFactory<C> cfac, GenPolynomial<C> A, C a)` Evaluate at main variable.
`static <C extends RingElem<C>> java.util.List<C>`	`evaluateMain(RingFactory<C> cfac, java.util.List<GenPolynomial<C>> L, C a)` Evaluate at main variable.
`static <C extends RingElem<C>> GenPolynomial<C>`	`evaluateMainRecursive(GenPolynomialRing<C> cfac, GenPolynomial<GenPolynomial<C>> A, C a)` Evaluate at main variable.
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`extendCoefficients(GenPolynomialRing<GenPolynomial<C>> pfac, GenPolynomial<GenPolynomial<C>> A, int j, long k)` Extend coefficient variables.
`static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>`	`extendCoefficients(GenSolvablePolynomialRing<GenPolynomial<C>> pfac, GenSolvablePolynomial<GenPolynomial<C>> A, int j, long k)` Extend coefficient variables.
`static BigInteger`	`factorBound(ExpVector e)` Factor coefficient bound.
`static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>>`	`fromAlgebraicCoefficients(GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomial<AlgebraicNumber<C>> A)` From AlgebraicNumber coefficients.
`static <C extends RingElem<C>> GenPolynomial<C>`	`fromIntegerCoefficients(GenPolynomialRing<C> fac, GenPolynomial<BigInteger> A)` From BigInteger coefficients.
`static <C extends RingElem<C>> java.util.List<GenPolynomial<C>>`	`fromIntegerCoefficients(GenPolynomialRing<C> fac, java.util.List<GenPolynomial<BigInteger>> L)` From BigInteger coefficients.
`static GenPolynomial<BigRational>`	`imaginaryPart(GenPolynomialRing<BigRational> fac, GenPolynomial<BigComplex> A)` Imaginary part.
`static <C extends RingElem<C>> GenPolynomial<C>`	`imaginaryPartFromComplex(GenPolynomialRing<C> fac, GenPolynomial<Complex<C>> A)` Imaginary part.
`static <C extends RingElem<C> & Modular> GenPolynomial<BigInteger>`	`integerFromModularCoefficients(GenPolynomialRing<BigInteger> fac, GenPolynomial<C> A)` BigInteger from ModInteger coefficients, symmetric.
`static <C extends RingElem<C> & Modular> java.util.List<GenPolynomial<BigInteger>>`	`integerFromModularCoefficients(GenPolynomialRing<BigInteger> fac, java.util.List<GenPolynomial<C>> L)` BigInteger from ModInteger coefficients, symmetric.
`static <C extends RingElem<C> & Modular> GenPolynomial<BigInteger>`	`integerFromModularCoefficientsPositive(GenPolynomialRing<BigInteger> fac, GenPolynomial<C> A)` BigInteger from ModInteger coefficients, positive.
`static GenPolynomial<BigInteger>`	`integerFromRationalCoefficients(GenPolynomialRing<BigInteger> fac, GenPolynomial<BigRational> A)` BigInteger from BigRational coefficients.
`static java.util.List<GenPolynomial<BigInteger>>`	`integerFromRationalCoefficients(GenPolynomialRing<BigInteger> fac, java.util.List<GenPolynomial<BigRational>> L)` BigInteger from BigRational coefficients.
`static java.lang.Object[]`	`integerFromRationalCoefficientsFactor(GenPolynomialRing<BigInteger> fac, GenPolynomial<BigRational> A)` BigInteger from BigRational coefficients.
`static <C extends RingElem<C>> GenPolynomial<C>`	`interpolate(GenPolynomialRing<C> fac, GenPolynomial<C> A, GenPolynomial<C> M, C mi, C a, C am)` Univariate polynomial interpolation.
`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)` ModInteger interpolate on first variable.
`static <C extends RingElem<C>> java.util.List<GenPolynomial<C>>`	`intersect(GenPolynomialRing<C> R, java.util.List<GenPolynomial<C>> F)` Intersection.
`static <C extends RingElem<C>> java.util.List<GenSolvablePolynomial<C>>`	`intersect(GenSolvablePolynomialRing<C> R, java.util.List<GenSolvablePolynomial<C>> F)` Intersection.
`static <C extends RingElem<C>> boolean`	`isBasePseudoQuotientRemainder(GenPolynomial<C> P, GenPolynomial<C> S, GenPolynomial<C> q, GenPolynomial<C> r)` Is GenPolynomial pseudo quotient and remainder.
`static <C extends RingElem<C>> boolean`	`isRecursivePseudoQuotientRemainder(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S, GenPolynomial<GenPolynomial<C>> q, GenPolynomial<GenPolynomial<C>> r)` Is recursive GenPolynomial pseudo quotient and remainder.
`static <C extends RingElem<C>> java.util.List<ExpVector>`	`leadingExpVector(java.util.List<GenPolynomial<C>> L)` Polynomial list leading exponent vectors.
`static <C extends RingElem<C>,D extends RingElem<D>> GenPolynomial<D>`	`map(GenPolynomialRing<D> ring, GenPolynomial<C> p, UnaryFunctor<C,D> f)` Map a unary function to the coefficients.
`static <C extends RingElem<C>> long`	`maxDegree(java.util.List<GenPolynomial<C>> P)` Maximal degree of polynomial list.
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`monic(GenPolynomial<GenPolynomial<C>> p)` GenPolynomial monic, i.e. leadingBaseCoefficient == 1.
`static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>`	`monic(GenSolvablePolynomial<GenPolynomial<C>> p)` GenSolvablePolynomial monic, i.e. leadingBaseCoefficient == 1.
`static <C extends RingElem<C>> java.util.List<GenPolynomial<C>>`	`monic(java.util.List<GenPolynomial<C>> L)` Polynomial list monic.
`static <C extends RingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>>`	`monicRec(java.util.List<GenPolynomial<GenPolynomial<C>>> L)` Recursive polynomial list monic.
`static GenPolynomial<BigRational>`	`realPart(GenPolynomialRing<BigRational> fac, GenPolynomial<BigComplex> A)` Real part.
`static <C extends RingElem<C>> GenPolynomial<C>`	`realPartFromComplex(GenPolynomialRing<C> fac, GenPolynomial<Complex<C>> A)` Real part.
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`recursive(GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomial<C> A)` Recursive representation.
`static <C extends RingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>>`	`recursive(GenPolynomialRing<GenPolynomial<C>> rfac, java.util.List<GenPolynomial<C>> L)` Recursive representation.
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`recursiveDensePseudoRemainder(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S)` GenPolynomial dense pseudo remainder.
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`recursiveDeriviative(GenPolynomial<GenPolynomial<C>> P)` GenPolynomial recursive polynomial derivative main variable.
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`recursiveDivide(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<C> s)` GenPolynomial divide.
`static <C extends RingElem<C>> GenWordPolynomial<GenPolynomial<C>>`	`recursiveDivide(GenWordPolynomial<GenPolynomial<C>> P, GenPolynomial<C> s)` GenPolynomial divide.
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`recursivePseudoDivide(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S)` GenPolynomial recursive pseudo divide.
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`recursivePseudoRemainder(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S)` Deprecated. Use `recursiveSparsePseudoRemainder(edu.jas.poly.GenPolynomial,edu.jas.poly.GenPolynomial)` instead
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`recursiveSparsePseudoRemainder(GenPolynomial<GenPolynomial<C>> P, GenPolynomial<GenPolynomial<C>> S)` GenPolynomial sparse pseudo remainder.
`static <C extends RingElem<C>> GenPolynomial<C>`	`removeUnusedLowerVariables(GenPolynomial<C> p)` Remove all lower variables which do not occur in polynomial.
`static <C extends RingElem<C>> GenPolynomial<C>`	`removeUnusedMiddleVariables(GenPolynomial<C> p)` Remove upper block of middle variables which do not occur in polynomial.
`static <C extends RingElem<C>> GenPolynomial<C>`	`removeUnusedUpperVariables(GenPolynomial<C> p)` Remove all upper variables which do not occur in polynomial.
`static <C extends RingElem<C>> GenPolynomial<C>`	`selectWithVariable(java.util.List<GenPolynomial<C>> P, int i)` Select polynomial with univariate leading term in variable i.
`static <C extends RingElem<C>> GenPolynomial<C>`	`seriesOfTaylor(GenPolynomial<C> f, C a)` Taylor series for polynomial.
`static <C extends RingElem<C>> GenPolynomial<C>`	`substituteMain(GenPolynomial<C> A, GenPolynomial<C> s)` Substitute main variable.
`static <C extends RingElem<C>> GenPolynomial<C>`	`substituteUnivariate(GenPolynomial<C> f, GenPolynomial<C> t)` Substitute univariate polynomial.
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`switchVariables(GenPolynomial<GenPolynomial<C>> P)` Recursive GenPolynomial switch varaible blocks.
`static <C extends RingElem<C>> GenPolynomial<Complex<C>>`	`toComplex(GenPolynomialRing<Complex<C>> fac, GenPolynomial<C> A)` Complex from real polynomial.
`static GenPolynomial<Product<ModInteger>>`	`toProduct(GenPolynomialRing<Product<ModInteger>> pfac, GenPolynomial<BigInteger> A)` Product representation.
`static java.util.List<GenPolynomial<Product<ModInteger>>>`	`toProduct(GenPolynomialRing<Product<ModInteger>> pfac, java.util.List<GenPolynomial<BigInteger>> L)` Product representation.
`static <C extends RingElem<C>> Product<GenPolynomial<C>>`	`toProduct(ProductRing<GenPolynomial<C>> pfac, C c, ExpVector e)` Product representation.
`static <C extends RingElem<C>> Product<GenPolynomial<C>>`	`toProduct(ProductRing<GenPolynomial<C>> pfac, GenPolynomial<C> A)` Product representation.
`static Product<ModInteger>`	`toProduct(ProductRing<ModInteger> pfac, BigInteger c)` Product representation.
`static <C extends GcdRingElem<C>> GenPolynomial<Product<C>>`	`toProductGen(GenPolynomialRing<Product<C>> pfac, GenPolynomial<C> A)` Product representation.
`static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<Product<C>>>`	`toProductGen(GenPolynomialRing<Product<C>> pfac, java.util.List<GenPolynomial<C>> L)` Product representation.
`static <C extends GcdRingElem<C>> Product<C>`	`toProductGen(ProductRing<C> pfac, C c)` Product representation.
`static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>>`	`toRecursive(GenPolynomialRing<GenPolynomial<C>> rfac, GenPolynomial<C> A)` To recursive representation.
`static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>>`	`toRecursive(GenSolvablePolynomialRing<GenPolynomial<C>> rfac, GenSolvablePolynomial<C> A)` To recursive representation.
`static <C extends RingElem<C>> long`	`totalDegree(java.util.List<GenPolynomial<C>> P)` Total degree of polynomial list.
`static <C extends RingElem<C>> long`	`totalDegreeLeadingTerm(java.util.List<GenPolynomial<C>> P)` Maximal degree of leading terms of a polynomial list.
`static <C extends RingElem<C>> java.util.List<GenWordPolynomial<C>>`	`wordMonic(java.util.List<GenWordPolynomial<C>> L)` Word polynomial list monic.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail
- PolyUtil
```
public PolyUtil()
```

Method Detail

recursive
```
public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursive(GenPolynomialRing<GenPolynomial<C>> rfac,
                                                                                GenPolynomial<C> A)
```
Recursive representation. Represent as polynomial in i variables with coefficients in n-i variables. Works for arbitrary term orders.

Type Parameters:

C - coefficient type.

Parameters:

rfac - recursive polynomial ring factory.

A - polynomial to be converted.

Returns:

Recursive represenations of this in the ring rfac.

distribute
```
public static <C extends RingElem<C>> GenPolynomial<C> distribute(GenPolynomialRing<C> dfac,
                                                                  GenPolynomial<GenPolynomial<C>> B)
```
Distribute a recursive polynomial to a generic polynomial. Works for arbitrary term orders.

Type Parameters:

C - coefficient type.

Parameters:

dfac - combined polynomial ring factory of coefficients and this.

B - polynomial to be converted.

Returns:

distributed polynomial.

recursive
```
public static <C extends RingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>> recursive(GenPolynomialRing<GenPolynomial<C>> rfac,
                                                                                                java.util.List<GenPolynomial<C>> L)
```
Recursive representation. Represent as polynomials in i variables with coefficients in n-i variables. Works for arbitrary term orders.

Type Parameters:

C - coefficient type.

Parameters:

rfac - recursive polynomial ring factory.

L - list of polynomials to be converted.

Returns:

Recursive represenations of the list in the ring rfac.

distribute
```
public static <C extends RingElem<C>> java.util.List<GenPolynomial<C>> distribute(GenPolynomialRing<C> dfac,
                                                                                  java.util.List<GenPolynomial<GenPolynomial<C>>> L)
```
Distribute a recursive polynomial list to a generic polynomial list. Works for arbitrary term orders.

Type Parameters:

C - coefficient type.

Parameters:

dfac - combined polynomial ring factory of coefficients and this.

L - list of polynomials to be converted.

Returns:

distributed polynomial list.

integerFromModularCoefficients

public static <C extends RingElem<C> & Modular> GenPolynomial<BigInteger> integerFromModularCoefficients(GenPolynomialRing<BigInteger> fac,
                                                                                                         GenPolynomial<C> A)

BigInteger from ModInteger coefficients, symmetric. Represent as polynomial with BigInteger coefficients by removing the modules and making coefficients symmetric to 0.

Parameters:: fac - result polynomial factory.; A - polynomial with ModInteger coefficients to be converted.
Returns:: polynomial with BigInteger coefficients.

integerFromModularCoefficients
```
public static <C extends RingElem<C> & Modular> java.util.List<GenPolynomial<BigInteger>> integerFromModularCoefficients(GenPolynomialRing<BigInteger> fac,
                                                                                                                         java.util.List<GenPolynomial<C>> L)
```
BigInteger from ModInteger coefficients, symmetric. Represent as polynomial with BigInteger coefficients by removing the modules and making coefficients symmetric to 0.

Parameters:

fac - result polynomial factory.

L - list of polynomials with ModInteger coefficients to be converted.

Returns:

list of polynomials with BigInteger coefficients.

integerFromModularCoefficientsPositive

public static <C extends RingElem<C> & Modular> GenPolynomial<BigInteger> integerFromModularCoefficientsPositive(GenPolynomialRing<BigInteger> fac,
                                                                                                                 GenPolynomial<C> A)

BigInteger from ModInteger coefficients, positive. Represent as polynomial with BigInteger coefficients by removing the modules.

Parameters:: fac - result polynomial factory.; A - polynomial with ModInteger coefficients to be converted.
Returns:: polynomial with BigInteger coefficients.

integerFromRationalCoefficients
```
public static GenPolynomial<BigInteger> integerFromRationalCoefficients(GenPolynomialRing<BigInteger> fac,
                                                                        GenPolynomial<BigRational> A)
```
BigInteger from BigRational coefficients. Represent as polynomial with BigInteger coefficients by multiplication with the lcm of the numerators of the BigRational coefficients.

Parameters:

fac - result polynomial factory.

A - polynomial with BigRational coefficients to be converted.

Returns:

polynomial with BigInteger coefficients.

integerFromRationalCoefficientsFactor
```
public static java.lang.Object[] integerFromRationalCoefficientsFactor(GenPolynomialRing<BigInteger> fac,
                                                                       GenPolynomial<BigRational> A)
```
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.
Author: Axel Kramer

Parameters:

fac - result polynomial factory.

A - polynomial with BigRational coefficients to be converted.

Returns:

Object[] with 3 entries: [0]->gcd [1]->lcm and [2]->polynomial with BigInteger coefficients.

integerFromRationalCoefficients
```
public static java.util.List<GenPolynomial<BigInteger>> integerFromRationalCoefficients(GenPolynomialRing<BigInteger> fac,
                                                                                        java.util.List<GenPolynomial<BigRational>> L)
```
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.

Parameters:

fac - result polynomial factory.

L - list of polynomials with BigRational coefficients to be converted.

Returns:

polynomial list with BigInteger coefficients.

fromIntegerCoefficients
```
public static <C extends RingElem<C>> GenPolynomial<C> fromIntegerCoefficients(GenPolynomialRing<C> fac,
                                                                               GenPolynomial<BigInteger> A)
```
From BigInteger coefficients. Represent as polynomial with type C coefficients, e.g. ModInteger or BigRational.

Type Parameters:

C - coefficient type.

Parameters:

fac - result polynomial factory.

A - polynomial with BigInteger coefficients to be converted.

Returns:

polynomial with type C coefficients.

fromIntegerCoefficients
```
public static <C extends RingElem<C>> java.util.List<GenPolynomial<C>> fromIntegerCoefficients(GenPolynomialRing<C> fac,
                                                                                               java.util.List<GenPolynomial<BigInteger>> L)
```
From BigInteger coefficients. Represent as list of polynomials with type C coefficients, e.g. ModInteger or BigRational.

Type Parameters:

C - coefficient type.

Parameters:

fac - result polynomial factory.

L - list of polynomials with BigInteger coefficients to be converted.

Returns:

list of polynomials with type C coefficients.

decimalFromRational

public static <C extends RingElem<C> & Rational> GenPolynomial<BigDecimal> decimalFromRational(GenPolynomialRing<BigDecimal> fac,
                                                                                               GenPolynomial<C> A)

Convert to decimal coefficients.

Parameters:: fac - result polynomial factory.; A - polynomial with Rational coefficients to be converted.
Returns:: polynomial with BigDecimal coefficients.

complexDecimalFromRational

public static <C extends RingElem<C> & Rational> GenPolynomial<Complex<BigDecimal>> complexDecimalFromRational(GenPolynomialRing<Complex<BigDecimal>> fac,
                                                                                                               GenPolynomial<Complex<C>> A)

Convert to complex decimal coefficients.

Parameters:: fac - result polynomial factory.; A - polynomial with complex Rational coefficients to be converted.
Returns:: polynomial with Complex BigDecimal coefficients.

realPart

public static GenPolynomial<BigRational> realPart(GenPolynomialRing<BigRational> fac,
                                                  GenPolynomial<BigComplex> A)

Real part.

Parameters:: fac - result polynomial factory.; A - polynomial with BigComplex coefficients to be converted.
Returns:: polynomial with real part of the coefficients.

imaginaryPart

public static GenPolynomial<BigRational> imaginaryPart(GenPolynomialRing<BigRational> fac,
                                                       GenPolynomial<BigComplex> A)

Imaginary part.

Parameters:: fac - result polynomial factory.; A - polynomial with BigComplex coefficients to be converted.
Returns:: polynomial with imaginary part of coefficients.

realPartFromComplex

public static <C extends RingElem<C>> GenPolynomial<C> realPartFromComplex(GenPolynomialRing<C> fac,
                                                                           GenPolynomial<Complex<C>> A)

Real part.

Parameters:: fac - result polynomial factory.; A - polynomial with BigComplex coefficients to be converted.
Returns:: polynomial with real part of the coefficients.

imaginaryPartFromComplex

public static <C extends RingElem<C>> GenPolynomial<C> imaginaryPartFromComplex(GenPolynomialRing<C> fac,
                                                                                GenPolynomial<Complex<C>> A)

Imaginary part.

Parameters:: fac - result polynomial factory.; A - polynomial with BigComplex coefficients to be converted.
Returns:: polynomial with imaginary part of coefficients.

toComplex

public static <C extends RingElem<C>> GenPolynomial<Complex<C>> toComplex(GenPolynomialRing<Complex<C>> fac,
                                                                          GenPolynomial<C> A)

Complex from real polynomial.

Parameters:: fac - result polynomial factory.; A - polynomial with C coefficients to be converted.
Returns:: polynomial with Complex coefficients.

complexFromRational

public static GenPolynomial<BigComplex> complexFromRational(GenPolynomialRing<BigComplex> fac,
                                                            GenPolynomial<BigRational> A)

Complex from rational coefficients.

Parameters:: fac - result polynomial factory.; A - polynomial with BigRational coefficients to be converted.
Returns:: polynomial with BigComplex coefficients.

complexFromAny

public static <C extends GcdRingElem<C>> GenPolynomial<Complex<C>> complexFromAny(GenPolynomialRing<Complex<C>> fac,
                                                                                  GenPolynomial<C> A)

Complex from ring element coefficients.

Parameters:: fac - result polynomial factory.; A - polynomial with RingElem coefficients to be converted.
Returns:: polynomial with Complex coefficients.

fromAlgebraicCoefficients

public static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>> fromAlgebraicCoefficients(GenPolynomialRing<GenPolynomial<C>> rfac,
                                                                                                   GenPolynomial<AlgebraicNumber<C>> A)

From AlgebraicNumber coefficients. Represent as polynomial with type GenPolynomial<C> coefficients, e.g. ModInteger or BigRational.

Parameters:: rfac - result polynomial factory.; A - polynomial with AlgebraicNumber coefficients to be converted.
Returns:: polynomial with type GenPolynomial<C> coefficients.

convertToAlgebraicCoefficients

public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> convertToAlgebraicCoefficients(GenPolynomialRing<AlgebraicNumber<C>> pfac,
                                                                                                          GenPolynomial<C> A)

Convert to AlgebraicNumber coefficients. Represent as polynomial with AlgebraicNumber coefficients, C is e.g. ModInteger or BigRational.

Parameters:: pfac - result polynomial factory.; A - polynomial with C coefficients to be converted.
Returns:: polynomial with AlgebraicNumber<C> coefficients.

convertToRecAlgebraicCoefficients

public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> convertToRecAlgebraicCoefficients(int depth,
                                                                                                             GenPolynomialRing<AlgebraicNumber<C>> pfac,
                                                                                                             GenPolynomial<C> A)

Convert to recursive AlgebraicNumber coefficients. Represent as polynomial with recursive AlgebraicNumber coefficients, C is e.g. ModInteger or BigRational.

Parameters:: depth - recursion depth of AlgebraicNumber coefficients.; pfac - result polynomial factory.; A - polynomial with C coefficients to be converted.
Returns:: polynomial with AlgebraicNumber<C> coefficients.

convertRecursiveToAlgebraicCoefficients

public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> convertRecursiveToAlgebraicCoefficients(GenPolynomialRing<AlgebraicNumber<C>> pfac,
                                                                                                                   GenPolynomial<GenPolynomial<C>> A)

Convert to AlgebraicNumber coefficients. Represent as polynomial with AlgebraicNumber coefficients, C is e.g. ModInteger or BigRational.

Parameters:: pfac - result polynomial factory.; A - recursive polynomial with GenPolynomial<BigInteger> coefficients to be converted.
Returns:: polynomial with AlgebraicNumber<C> coefficients.

complexFromAlgebraic

public static <C extends GcdRingElem<C>> GenPolynomial<Complex<C>> complexFromAlgebraic(GenPolynomialRing<Complex<C>> fac,
                                                                                        GenPolynomial<AlgebraicNumber<C>> A)

Complex from algebraic coefficients.

Parameters:: fac - result polynomial factory.; A - polynomial with AlgebraicNumber coefficients Q(i) to be converted.
Returns:: polynomial with Complex coefficients.

algebraicFromComplex

public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> algebraicFromComplex(GenPolynomialRing<AlgebraicNumber<C>> fac,
                                                                                                GenPolynomial<Complex<C>> A)

AlgebraicNumber from complex coefficients.

Parameters:: fac - result polynomial factory over Q(i).; A - polynomial with Complex coefficients to be converted.
Returns:: polynomial with AlgebraicNumber coefficients.

chineseRemainder

public static <C extends RingElem<C> & Modular> GenPolynomial<C> chineseRemainder(GenPolynomialRing<C> fac,
                                                                                  GenPolynomial<C> A,
                                                                                  C mi,
                                                                                  GenPolynomial<C> B)

ModInteger chinese remainder algorithm on coefficients.

Parameters:: fac - GenPolynomial<ModInteger> result factory with A.coFac.modul*B.coFac.modul = C.coFac.modul.; A - GenPolynomial<ModInteger>.; B - other GenPolynomial<ModInteger>.; mi - inverse of A.coFac.modul in ring B.coFac.
Returns:: S = cra(A,B), with S mod A.coFac.modul == A and S mod B.coFac.modul == B.

monic
```
public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> monic(GenPolynomial<GenPolynomial<C>> p)
```
GenPolynomial monic, i.e. leadingBaseCoefficient == 1. If leadingBaseCoefficient is not invertible returns this unmodified.

Type Parameters:

C - coefficient type.

Parameters:

p - recursive GenPolynomial>.

Returns:

monic(p).

monic
```
public static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> monic(GenSolvablePolynomial<GenPolynomial<C>> p)
```
GenSolvablePolynomial monic, i.e. leadingBaseCoefficient == 1. If leadingBaseCoefficient is not invertible returns this unmodified.

Type Parameters:

C - coefficient type.

Parameters:

p - recursive GenSolvablePolynomial>.

Returns:

monic(p).

monic
```
public static <C extends RingElem<C>> java.util.List<GenPolynomial<C>> monic(java.util.List<GenPolynomial<C>> L)
```
Polynomial list monic.

Type Parameters:

C - coefficient type.

Parameters:

L - list of polynomials with field coefficients.

Returns:

list of polynomials with leading coefficient 1.

wordMonic
```
public static <C extends RingElem<C>> java.util.List<GenWordPolynomial<C>> wordMonic(java.util.List<GenWordPolynomial<C>> L)
```
Word polynomial list monic.

Type Parameters:

C - coefficient type.

Parameters:

L - list of word polynomials with field coefficients.

Returns:

list of word polynomials with leading coefficient 1.

monicRec
```
public static <C extends RingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>> monicRec(java.util.List<GenPolynomial<GenPolynomial<C>>> L)
```
Recursive polynomial list monic.

Type Parameters:

C - coefficient type.

Parameters:

L - list of recursive polynomials with field coefficients.

Returns:

list of polynomials with leading base coefficient 1.

leadingExpVector
```
public static <C extends RingElem<C>> java.util.List<ExpVector> leadingExpVector(java.util.List<GenPolynomial<C>> L)
```
Polynomial list leading exponent vectors.

Type Parameters:

C - coefficient type.

Parameters:

L - list of polynomials.

Returns:

list of leading exponent vectors.

extendCoefficients

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> extendCoefficients(GenPolynomialRing<GenPolynomial<C>> pfac,
                                                                                         GenPolynomial<GenPolynomial<C>> A,
                                                                                         int j,
                                                                                         long k)

Extend coefficient variables. Extend all coefficient ExpVectors by i elements and multiply by x_j^k.

Parameters:: pfac - extended polynomial ring factory (by i variables in the coefficients).; j - index of variable to be used for multiplication.; k - exponent for x_j.
Returns:: extended polynomial.

extendCoefficients

public static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> extendCoefficients(GenSolvablePolynomialRing<GenPolynomial<C>> pfac,
                                                                                                 GenSolvablePolynomial<GenPolynomial<C>> A,
                                                                                                 int j,
                                                                                                 long k)

Extend coefficient variables. Extend all coefficient ExpVectors by i elements and multiply by x_j^k.

Parameters:: pfac - extended polynomial ring factory (by i variables in the coefficients).; j - index of variable to be used for multiplication.; k - exponent for x_j.
Returns:: extended polynomial.

toRecursive
```
public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> toRecursive(GenPolynomialRing<GenPolynomial<C>> rfac,
                                                                                  GenPolynomial<C> A)
```
To recursive representation. Represent as polynomial in i+r variables with coefficients in i variables. Works for arbitrary term orders.

Type Parameters:

C - coefficient type.

Parameters:

rfac - recursive polynomial ring factory.

A - polynomial to be converted.

Returns:

Recursive represenations of A in the ring rfac.

toRecursive

public static <C extends RingElem<C>> GenSolvablePolynomial<GenPolynomial<C>> toRecursive(GenSolvablePolynomialRing<GenPolynomial<C>> rfac,
                                                                                          GenSolvablePolynomial<C> A)

To recursive representation. Represent as solvable polynomial in i+r variables with coefficients in i variables. Works for arbitrary term orders.

Type Parameters:: C - coefficient type.
Parameters:: rfac - recursive solvable polynomial ring factory.; A - solvable polynomial to be converted.
Returns:: Recursive represenations of A in the ring rfac.

baseRemainderPoly
```
public static <C extends RingElem<C>> GenPolynomial<C> baseRemainderPoly(GenPolynomial<C> P,
                                                                         C s)
```
GenPolynomial coefficient wise remainder.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

s - nonzero coefficient.

Returns:

coefficient wise remainder.

See Also:

GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

basePseudoRemainder
```
@Deprecated
public static <C extends RingElem<C>> GenPolynomial<C> basePseudoRemainder(GenPolynomial<C> P,
                                                                                       GenPolynomial<C> S)
```
Deprecated. Use baseSparsePseudoRemainder(edu.jas.poly.GenPolynomial,edu.jas.poly.GenPolynomial) instead

GenPolynomial sparse pseudo remainder. For univariate polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

S - nonzero GenPolynomial.

Returns:

remainder with ldcf(S)^m' P = quotient * S + remainder. m' ≤ deg(P)-deg(S)

See Also:

GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

baseSparsePseudoRemainder
```
public static <C extends RingElem<C>> GenPolynomial<C> baseSparsePseudoRemainder(GenPolynomial<C> P,
                                                                                 GenPolynomial<C> S)
```
GenPolynomial sparse pseudo remainder. For univariate polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

S - nonzero GenPolynomial.

Returns:

remainder with ldcf(S)^m' P = quotient * S + remainder. m' ≤ deg(P)-deg(S)

See Also:

GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

baseDensePseudoRemainder
```
public static <C extends RingElem<C>> GenPolynomial<C> baseDensePseudoRemainder(GenPolynomial<C> P,
                                                                                GenPolynomial<C> S)
```
GenPolynomial dense pseudo remainder. For univariate polynomials.

Parameters:

P - GenPolynomial.

S - nonzero GenPolynomial.

Returns:

remainder with ldcf(S)^m P = quotient * S + remainder. m == deg(P)-deg(S)

See Also:

GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

baseDensePseudoQuotient
```
public static <C extends RingElem<C>> GenPolynomial<C> baseDensePseudoQuotient(GenPolynomial<C> P,
                                                                               GenPolynomial<C> S)
```
GenPolynomial dense pseudo quotient. For univariate polynomials.

Parameters:

P - GenPolynomial.

S - nonzero GenPolynomial.

Returns:

quotient with ldcf(S)^m P = quotient * S + remainder. m == deg(P)-deg(S)

See Also:

GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

basePseudoDivide
```
public static <C extends RingElem<C>> GenPolynomial<C> basePseudoDivide(GenPolynomial<C> P,
                                                                        GenPolynomial<C> S)
```
GenPolynomial sparse pseudo divide. For univariate polynomials or exact division.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

S - nonzero GenPolynomial.

Returns:

quotient with ldcf(S)^m' P = quotient * S + remainder. m' ≤ deg(P)-deg(S)

See Also:

GenPolynomial.divide(edu.jas.poly.GenPolynomial).

basePseudoQuotientRemainder
```
public static <C extends RingElem<C>> GenPolynomial<C>[] basePseudoQuotientRemainder(GenPolynomial<C> P,
                                                                                     GenPolynomial<C> S)
```
GenPolynomial sparse pseudo quotient and remainder. For univariate polynomials or exact division.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

S - nonzero GenPolynomial.

Returns:

[ quotient, remainder ] with ldcf(S)^m' P = quotient * S + remainder. m' ≤ deg(P)-deg(S)

See Also:

GenPolynomial.divide(edu.jas.poly.GenPolynomial).

isBasePseudoQuotientRemainder

public static <C extends RingElem<C>> boolean isBasePseudoQuotientRemainder(GenPolynomial<C> P,
                                                                            GenPolynomial<C> S,
                                                                            GenPolynomial<C> q,
                                                                            GenPolynomial<C> r)

Is GenPolynomial pseudo quotient and remainder. For univariate polynomials.

Type Parameters:: C - coefficient type.
Parameters:: P - base GenPolynomial.; S - nonzero base GenPolynomial.
Returns:: true, if P = q * S + r, else false.
See Also:: Note: not always meaningful and working

recursiveDivide

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursiveDivide(GenPolynomial<GenPolynomial<C>> P,
                                                                                      GenPolynomial<C> s)

GenPolynomial divide. For recursive polynomials. Division by coefficient ring element.

Type Parameters:: C - coefficient type.
Parameters:: P - recursive GenPolynomial.; s - GenPolynomial.
Returns:: this/s.

recursiveDivide

public static <C extends RingElem<C>> GenWordPolynomial<GenPolynomial<C>> recursiveDivide(GenWordPolynomial<GenPolynomial<C>> P,
                                                                                          GenPolynomial<C> s)

GenPolynomial divide. For recursive polynomials. Division by coefficient ring element.

Type Parameters:: C - coefficient type.
Parameters:: P - recursive GenPolynomial.; s - GenPolynomial.
Returns:: this/s.

baseRecursiveDivide

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> baseRecursiveDivide(GenPolynomial<GenPolynomial<C>> P,
                                                                                          C s)

GenPolynomial base divide. For recursive polynomials. Division by coefficient ring element.

Type Parameters:: C - coefficient type.
Parameters:: P - recursive GenPolynomial.; s - coefficient.
Returns:: this/s.

recursivePseudoRemainder
```
@Deprecated
public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursivePseudoRemainder(GenPolynomial<GenPolynomial<C>> P,
                                                                                                           GenPolynomial<GenPolynomial<C>> S)
```
Deprecated. Use recursiveSparsePseudoRemainder(edu.jas.poly.GenPolynomial,edu.jas.poly.GenPolynomial) instead

GenPolynomial sparse pseudo remainder. For recursive polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial.

S - nonzero recursive GenPolynomial.

Returns:

remainder with ldcf(S)^m' P = quotient * S + remainder.

See Also:

GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

recursiveSparsePseudoRemainder

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursiveSparsePseudoRemainder(GenPolynomial<GenPolynomial<C>> P,
                                                                                                     GenPolynomial<GenPolynomial<C>> S)

GenPolynomial sparse pseudo remainder. For recursive polynomials.

Type Parameters:: C - coefficient type.
Parameters:: P - recursive GenPolynomial.; S - nonzero recursive GenPolynomial.
Returns:: remainder with ldcf(S)^m' P = quotient * S + remainder.
See Also:: GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

recursiveDensePseudoRemainder

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursiveDensePseudoRemainder(GenPolynomial<GenPolynomial<C>> P,
                                                                                                    GenPolynomial<GenPolynomial<C>> S)

GenPolynomial dense pseudo remainder. For recursive polynomials.

Parameters:: P - recursive GenPolynomial.; S - nonzero recursive GenPolynomial.
Returns:: remainder with ldcf(S)^m' P = quotient * S + remainder.
See Also:: GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

recursivePseudoDivide

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursivePseudoDivide(GenPolynomial<GenPolynomial<C>> P,
                                                                                            GenPolynomial<GenPolynomial<C>> S)

GenPolynomial recursive pseudo divide. For recursive polynomials.

Type Parameters:: C - coefficient type.
Parameters:: P - recursive GenPolynomial.; S - nonzero recursive GenPolynomial.
Returns:: quotient with ldcf(S)^m' P = quotient * S + remainder.
See Also:: GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

isRecursivePseudoQuotientRemainder

public static <C extends RingElem<C>> boolean isRecursivePseudoQuotientRemainder(GenPolynomial<GenPolynomial<C>> P,
                                                                                 GenPolynomial<GenPolynomial<C>> S,
                                                                                 GenPolynomial<GenPolynomial<C>> q,
                                                                                 GenPolynomial<GenPolynomial<C>> r)

Is recursive GenPolynomial pseudo quotient and remainder. For recursive polynomials.

Type Parameters:: C - coefficient type.
Parameters:: P - recursive GenPolynomial.; S - nonzero recursive GenPolynomial.
Returns:: true, if P ~= q * S + r, else false.
See Also:: Note: not always meaningful and working

coefficientPseudoDivide

public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> coefficientPseudoDivide(GenPolynomial<GenPolynomial<C>> P,
                                                                                              GenPolynomial<C> s)

GenPolynomial pseudo divide. For recursive polynomials.

Type Parameters:: C - coefficient type.
Parameters:: P - recursive GenPolynomial.; s - nonzero GenPolynomial.
Returns:: quotient with ldcf(s)^m P = quotient * s + remainder.
See Also:: GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

coefficientBasePseudoDivide
```
public static <C extends RingElem<C>> GenPolynomial<C> coefficientBasePseudoDivide(GenPolynomial<C> P,
                                                                                   C s)
```
GenPolynomial pseudo divide. For polynomials.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

s - nonzero coefficient.

Returns:

quotient with ldcf(s)^m P = quotient * s + remainder.

See Also:

GenPolynomial.remainder(edu.jas.poly.GenPolynomial).

baseDeriviative
```
public static <C extends RingElem<C>> GenPolynomial<C> baseDeriviative(GenPolynomial<C> P)
```
GenPolynomial polynomial derivative main variable.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

Returns:

deriviative(P).

baseDeriviative
```
public static <C extends RingElem<C>> GenPolynomial<C> baseDeriviative(GenPolynomial<C> P,
                                                                       int r)
```
GenPolynomial polynomial partial derivative variable r.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

r - variable for partial deriviate.

Returns:

deriviative(P,r).

baseIntegral
```
public static <C extends RingElem<C>> GenPolynomial<C> baseIntegral(GenPolynomial<C> P)
```
GenPolynomial polynomial integral main variable.

Type Parameters:

C - coefficient type.

Parameters:

P - GenPolynomial.

Returns:

integral(P).

recursiveDeriviative
```
public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> recursiveDeriviative(GenPolynomial<GenPolynomial<C>> P)
```
GenPolynomial recursive polynomial derivative main variable.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial.

Returns:

deriviative(P).

factorBound
```
public static BigInteger factorBound(ExpVector e)
```
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.

Parameters:

e - degree vector of a GenPolynomial A.

Returns:

2**b.

evaluateMainRecursive

public static <C extends RingElem<C>> GenPolynomial<C> evaluateMainRecursive(GenPolynomialRing<C> cfac,
                                                                             GenPolynomial<GenPolynomial<C>> A,
                                                                             C a)

Evaluate at main variable.

Type Parameters:: C - coefficient type.
Parameters:: cfac - coefficent polynomial ring factory.; A - recursive polynomial to be evaluated.; a - value to evaluate at.
Returns:: A( x_1, ..., x_{n-1}, a ).

evaluateMain

public static <C extends RingElem<C>> GenPolynomial<C> evaluateMain(GenPolynomialRing<C> cfac,
                                                                    GenPolynomial<C> A,
                                                                    C a)

Evaluate at main variable.

Type Parameters:: C - coefficient type.
Parameters:: cfac - coefficent polynomial ring factory.; A - distributed polynomial to be evaluated.; a - value to evaluate at.
Returns:: A( x_1, ..., x_{n-1}, a ).

evaluateMain

public static <C extends RingElem<C>> java.util.List<GenPolynomial<C>> evaluateMain(GenPolynomialRing<C> cfac,
                                                                                    java.util.List<GenPolynomial<C>> L,
                                                                                    C a)

Evaluate at main variable.

Type Parameters:: C - coefficient type.
Parameters:: cfac - coefficent ring factory.; L - list of univariate polynomials to be evaluated.; a - value to evaluate at.
Returns:: list( A( x_1, ..., x_{n-1}, a ) ) for A in L.

evaluateMain

public static <C extends RingElem<C>> C evaluateMain(RingFactory<C> cfac,
                                                     GenPolynomial<C> A,
                                                     C a)

Evaluate at main variable.

Type Parameters:: C - coefficient type.
Parameters:: cfac - coefficent ring factory.; A - univariate polynomial to be evaluated.; a - value to evaluate at.
Returns:: A( a ).

evaluateMain

public static <C extends RingElem<C>> java.util.List<C> evaluateMain(RingFactory<C> cfac,
                                                                     java.util.List<GenPolynomial<C>> L,
                                                                     C a)

Evaluate at main variable.

Type Parameters:: C - coefficient type.
Parameters:: cfac - coefficent ring factory.; L - list of univariate polynomial to be evaluated.; a - value to evaluate at.
Returns:: list( A( a ) ) for A in L.

evaluate

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)

Evaluate at k-th variable.

Type Parameters:: C - coefficient type.
Parameters:: cfac - coefficient polynomial ring in k variables C[x_1, ..., x_k] factory.; 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.; 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.; dfac - polynomial ring in n-1 variables. C[x_1, ..., x_{k-1}, x_{k+1}, ..., x_n] factory.; A - polynomial to be evaluated.; a - value to evaluate at.
Returns:: A( x_1, ..., x_{k-1}, a, x_{k+1}, ..., x_n).

evaluateFirst

public static <C extends RingElem<C>> GenPolynomial<C> evaluateFirst(GenPolynomialRing<C> cfac,
                                                                     GenPolynomialRing<C> dfac,
                                                                     GenPolynomial<C> A,
                                                                     C a)

Evaluate at first (lowest) variable.

Type Parameters:: C - coefficient type.
Parameters:: cfac - coefficient polynomial ring in first variable C[x_1] factory.; dfac - polynomial ring in n-1 variables. C[x_2, ..., x_n] factory.; A - polynomial to be evaluated.; a - value to evaluate at.
Returns:: A( a, x_2, ..., x_n).

evaluateFirstRec

public static <C extends RingElem<C>> GenPolynomial<C> evaluateFirstRec(GenPolynomialRing<C> cfac,
                                                                        GenPolynomialRing<C> dfac,
                                                                        GenPolynomial<GenPolynomial<C>> A,
                                                                        C a)

Evaluate at first (lowest) variable.

Type Parameters:: C - coefficient type. Could also be called evaluateFirst(), but type erasure of A parameter does not allow same name.
Parameters:: cfac - coefficient polynomial ring in first variable C[x_1] factory.; dfac - polynomial ring in n-1 variables. C[x_2, ..., x_n] factory.; A - recursive polynomial to be evaluated.; a - value to evaluate at.
Returns:: A( a, x_2, ..., x_n).

evaluateAll

@Deprecated
public static <C extends RingElem<C>> C evaluateAll(RingFactory<C> cfac,
                                                                GenPolynomialRing<C> dfac,
                                                                GenPolynomial<C> A,
                                                                java.util.List<C> a)

Deprecated. use evaluateAll() with three arguments

Evaluate all variables.

Type Parameters:: C - coefficient type.
Parameters:: cfac - coefficient ring factory.; dfac - polynomial ring in n variables. C[x_1, x_2, ..., x_n] factory.; A - polynomial to be evaluated.; a - = (a_1, a_2, ..., a_n) a tuple of values to evaluate at.
Returns:: A(a_1, a_2, ..., a_n).

evaluateAll

public static <C extends RingElem<C>> C evaluateAll(RingFactory<C> cfac,
                                                    GenPolynomial<C> A,
                                                    java.util.List<C> a)

Evaluate all variables.

Type Parameters:: C - coefficient type.
Parameters:: cfac - coefficient ring factory.; A - polynomial to be evaluated.; a - = (a_1, a_2, ..., a_n) a tuple of values to evaluate at.
Returns:: A(a_1, a_2, ..., a_n).

substituteMain

public static <C extends RingElem<C>> GenPolynomial<C> substituteMain(GenPolynomial<C> A,
                                                                      GenPolynomial<C> s)

Substitute main variable.

Parameters:: A - univariate polynomial.; s - polynomial for substitution.
Returns:: polynomial A(x <- s).

substituteUnivariate

public static <C extends RingElem<C>> GenPolynomial<C> substituteUnivariate(GenPolynomial<C> f,
                                                                            GenPolynomial<C> t)

Substitute univariate polynomial.

Parameters:: f - univariate polynomial.; t - polynomial for substitution.
Returns:: polynomial A(x <- t).

seriesOfTaylor

public static <C extends RingElem<C>> GenPolynomial<C> seriesOfTaylor(GenPolynomial<C> f,
                                                                      C a)

Taylor series for polynomial.

Parameters:: f - univariate polynomial.; a - expansion point.
Returns:: Taylor series (a polynomial) of f at a.

interpolate

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)

ModInteger interpolate on first variable.

Type Parameters:: C - coefficient type.
Parameters:: fac - GenPolynomial result factory.; A - GenPolynomial.; M - GenPolynomial interpolation modul of A.; mi - inverse of M(am) in ring fac.coFac.; B - evaluation of other GenPolynomial.; am - evaluation point (interpolation modul) of B, i.e. P(am) = B.
Returns:: S, with S mod M == A and S(am) == B.

interpolate

public static <C extends RingElem<C>> GenPolynomial<C> interpolate(GenPolynomialRing<C> fac,
                                                                   GenPolynomial<C> A,
                                                                   GenPolynomial<C> M,
                                                                   C mi,
                                                                   C a,
                                                                   C am)

Univariate polynomial interpolation.

Type Parameters:: C - coefficient type.
Parameters:: fac - GenPolynomial result factory.; A - GenPolynomial.; M - GenPolynomial interpolation modul of A.; mi - inverse of M(am) in ring fac.coFac.; a - evaluation of other GenPolynomial.; am - evaluation point (interpolation modul) of a, i.e. P(am) = a.
Returns:: S, with S mod M == A and S(am) == a.

switchVariables
```
public static <C extends RingElem<C>> GenPolynomial<GenPolynomial<C>> switchVariables(GenPolynomial<GenPolynomial<C>> P)
```
Recursive GenPolynomial switch varaible blocks.

Type Parameters:

C - coefficient type.

Parameters:

P - recursive GenPolynomial in R[X,Y].

Returns:

this in R[Y,X].

totalDegreeLeadingTerm
```
public static <C extends RingElem<C>> long totalDegreeLeadingTerm(java.util.List<GenPolynomial<C>> P)
```
Maximal degree of leading terms of a polynomial list.

Returns:

maximum degree of the leading terms of a polynomial list.

totalDegree

public static <C extends RingElem<C>> long totalDegree(java.util.List<GenPolynomial<C>> P)

Total degree of polynomial list.

Returns:: total degree of the polynomial list.

maxDegree

public static <C extends RingElem<C>> long maxDegree(java.util.List<GenPolynomial<C>> P)

Maximal degree of polynomial list.

Returns:: maximal degree of the polynomial list.

coeffMaxDegree
```
public static <C extends RingElem<C>> long coeffMaxDegree(GenPolynomial<GenPolynomial<C>> A)
```
Maximal degree in the coefficient polynomials.

Type Parameters:

C - coefficient type.

Returns:

maximal degree in the coefficients.

map

public static <C extends RingElem<C>,D extends RingElem<D>> GenPolynomial<D> map(GenPolynomialRing<D> ring,
                                                                                 GenPolynomial<C> p,
                                                                                 UnaryFunctor<C,D> f)

Map a unary function to the coefficients.

Parameters:: ring - result polynomial ring factory.; p - polynomial.; f - evaluation functor.
Returns:: new polynomial with coefficients f(p(e)).

toProductGen

public static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<Product<C>>> toProductGen(GenPolynomialRing<Product<C>> pfac,
                                                                                                java.util.List<GenPolynomial<C>> L)

Product representation.

Type Parameters:: C - coefficient type.
Parameters:: pfac - polynomial ring factory.; L - list of polynomials to be represented.
Returns:: Product represenation of L in the polynomial ring pfac.

toProductGen

public static <C extends GcdRingElem<C>> GenPolynomial<Product<C>> toProductGen(GenPolynomialRing<Product<C>> pfac,
                                                                                GenPolynomial<C> A)

Product representation.

Type Parameters:: C - coefficient type.
Parameters:: pfac - polynomial ring factory.; A - polynomial to be represented.
Returns:: Product represenation of A in the polynomial ring pfac.

toProductGen
```
public static <C extends GcdRingElem<C>> Product<C> toProductGen(ProductRing<C> pfac,
                                                                 C c)
```
Product representation.

Type Parameters:

C - coefficient type.

Parameters:

pfac - product ring factory.

c - coefficient to be represented.

Returns:

Product represenation of c in the ring pfac.

toProduct

public static <C extends RingElem<C>> Product<GenPolynomial<C>> toProduct(ProductRing<GenPolynomial<C>> pfac,
                                                                          C c,
                                                                          ExpVector e)

Product representation.

Type Parameters:: C - coefficient type.
Parameters:: pfac - product polynomial ring factory.; c - coefficient to be used.; e - exponent vector.
Returns:: Product represenation of c X^e in the ring pfac.

toProduct

public static <C extends RingElem<C>> Product<GenPolynomial<C>> toProduct(ProductRing<GenPolynomial<C>> pfac,
                                                                          GenPolynomial<C> A)

Product representation.

Type Parameters:: C - coefficient type.
Parameters:: pfac - product polynomial ring factory.; A - polynomial.
Returns:: Product represenation of the terms of A in the ring pfac.

toProduct

public static Product<ModInteger> toProduct(ProductRing<ModInteger> pfac,
                                            BigInteger c)

Product representation.

Parameters:: pfac - product ring factory.; c - coefficient to be represented.
Returns:: Product represenation of c in the ring pfac.

toProduct

public static GenPolynomial<Product<ModInteger>> toProduct(GenPolynomialRing<Product<ModInteger>> pfac,
                                                           GenPolynomial<BigInteger> A)

Product representation.

Parameters:: pfac - polynomial ring factory.; A - polynomial to be represented.
Returns:: Product represenation of A in the polynomial ring pfac.

toProduct

public static java.util.List<GenPolynomial<Product<ModInteger>>> toProduct(GenPolynomialRing<Product<ModInteger>> pfac,
                                                                           java.util.List<GenPolynomial<BigInteger>> L)

Product representation.

Parameters:: pfac - polynomial ring factory.; L - list of polynomials to be represented.
Returns:: Product represenation of L in the polynomial ring pfac.

intersect
```
public static <C extends RingElem<C>> java.util.List<GenPolynomial<C>> intersect(GenPolynomialRing<C> R,
                                                                                 java.util.List<GenPolynomial<C>> F)
```
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.

Parameters:

R - polynomial ring

F - list of polynomials

Returns:

R \cap F

intersect
```
public static <C extends RingElem<C>> java.util.List<GenSolvablePolynomial<C>> intersect(GenSolvablePolynomialRing<C> R,
                                                                                         java.util.List<GenSolvablePolynomial<C>> F)
```
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.

Parameters:

R - solvable polynomial ring

F - list of solvable polynomials

Returns:

R \cap F

removeUnusedUpperVariables
```
public static <C extends RingElem<C>> GenPolynomial<C> removeUnusedUpperVariables(GenPolynomial<C> p)
```
Remove all upper variables which do not occur in polynomial.

Parameters:

p - polynomial.

Returns:

polynomial with removed variables

removeUnusedLowerVariables
```
public static <C extends RingElem<C>> GenPolynomial<C> removeUnusedLowerVariables(GenPolynomial<C> p)
```
Remove all lower variables which do not occur in polynomial.

Parameters:

p - polynomial.

Returns:

polynomial with removed variables

removeUnusedMiddleVariables
```
public static <C extends RingElem<C>> GenPolynomial<C> removeUnusedMiddleVariables(GenPolynomial<C> p)
```
Remove upper block of middle variables which do not occur in polynomial.

Parameters:

p - polynomial.

Returns:

polynomial with removed variables

selectWithVariable

public static <C extends RingElem<C>> GenPolynomial<C> selectWithVariable(java.util.List<GenPolynomial<C>> P,
                                                                          int i)

Select polynomial with univariate leading term in variable i.

Parameters:: i - variable index.
Returns:: polynomial with head term in variable i

Class PolyUtil

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

PolyUtil

Method Detail

recursive

distribute

recursive

distribute

integerFromModularCoefficients

integerFromModularCoefficients

integerFromModularCoefficientsPositive

integerFromRationalCoefficients

integerFromRationalCoefficientsFactor

integerFromRationalCoefficients

fromIntegerCoefficients

fromIntegerCoefficients

decimalFromRational

complexDecimalFromRational

realPart

imaginaryPart

realPartFromComplex

imaginaryPartFromComplex

toComplex

complexFromRational

complexFromAny

fromAlgebraicCoefficients

convertToAlgebraicCoefficients

convertToRecAlgebraicCoefficients

convertRecursiveToAlgebraicCoefficients

complexFromAlgebraic

algebraicFromComplex

chineseRemainder

monic

monic

monic

wordMonic

monicRec

leadingExpVector

extendCoefficients

extendCoefficients

toRecursive

toRecursive

baseRemainderPoly

basePseudoRemainder

baseSparsePseudoRemainder

baseDensePseudoRemainder

baseDensePseudoQuotient

basePseudoDivide

basePseudoQuotientRemainder

isBasePseudoQuotientRemainder

recursiveDivide

recursiveDivide

baseRecursiveDivide

recursivePseudoRemainder

recursiveSparsePseudoRemainder

recursiveDensePseudoRemainder

recursivePseudoDivide

isRecursivePseudoQuotientRemainder

coefficientPseudoDivide

coefficientBasePseudoDivide

baseDeriviative

baseDeriviative

baseIntegral

recursiveDeriviative

factorBound

evaluateMainRecursive

evaluateMain

evaluateMain

evaluateMain

evaluateMain

evaluate

evaluateFirst

evaluateFirstRec

evaluateAll

evaluateAll

substituteMain

substituteUnivariate