Class PolynomialAlgebra

java.lang.Object

org.episteme.core.mathematics.algebra.polynomials.PolynomialAlgebra

public class PolynomialAlgebra
extends Object

Advanced polynomial operations: GCD, factorization, resultants.

Algorithms for symbolic computation and computer algebra.

Since:

1.0

Author:

Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static class	PolynomialAlgebra.ExtendedGCDResult	Extended Euclidean algorithm: finds s, t such that gcd(a,b) = s*a + t*b

Constructor Summary

Constructors

Constructor	Description
PolynomialAlgebra()

Method Summary

Modifier and Type	Method	Description
static Integer	content(Polynomial<Integer> p)	Content of polynomial (GCD of all coefficients).
static Polynomial<Integer>	derivative(Polynomial<Integer> p)	Computes derivative of polynomial.
static Polynomial<Integer>	derivative(Polynomial<Integer> p, int n)	Computes nth derivative.
static Integer	discriminant(Polynomial<Integer> p)	Discriminant of polynomial.
static Polynomial<Integer>	gcd(Polynomial<Integer> a, Polynomial<Integer> b)	Euclidean algorithm for polynomial GCD.
static boolean	isIrreducible(Polynomial<Integer> p)	Checks if polynomial is irreducible.
static Polynomial<Integer>	primitivePart(Polynomial<Integer> p)	Primitive part: polynomial / content
static Integer	resultant(Polynomial<Integer> a, Polynomial<Integer> b)	Resultant of two polynomials.
static List<Polynomial<Integer>>	squareFreeFactorization(Polynomial<Integer> f)	Square-free factorization: f = ∏ f_i^i

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

PolynomialAlgebra

public PolynomialAlgebra()

Method Details

gcd

public static Polynomial<Integer> gcd(Polynomial<Integer> a,
 Polynomial<Integer> b)

Euclidean algorithm for polynomial GCD.

gcd(a, b) = gcd(b, a mod b) Works for polynomials over fields.

Parameters:

a - first polynomial

b - second polynomial

Returns:

greatest common divisor

resultant

public static Integer resultant(Polynomial<Integer> a,
 Polynomial<Integer> b)

Resultant of two polynomials.

Res(a, b) = 0 iff a and b have common root. Computed using Sylvester matrix determinant.

discriminant

public static Integer discriminant(Polynomial<Integer> p)

Discriminant of polynomial.

Δ(f) = (-1)^(n(n-1)/2) * Res(f, f') / lc(f) Zero iff f has repeated root.

squareFreeFactorization

public static List<Polynomial<Integer>> squareFreeFactorization(Polynomial<Integer> f)

Square-free factorization: f = ∏ f_i^i

Separates polynomial into square-free parts. First step in complete factorization.

content

public static Integer content(Polynomial<Integer> p)

Content of polynomial (GCD of all coefficients).

primitivePart

public static Polynomial<Integer> primitivePart(Polynomial<Integer> p)

Primitive part: polynomial / content

pp(f) * cont(f) = f

isIrreducible

public static boolean isIrreducible(Polynomial<Integer> p)

Checks if polynomial is irreducible.

Uses Eisenstein's criterion and other tests.

derivative

public static Polynomial<Integer> derivative(Polynomial<Integer> p)

Computes derivative of polynomial.

derivative

public static Polynomial<Integer> derivative(Polynomial<Integer> p,
 int n)

Computes nth derivative.