Multivariate Polynomial Rings
- Introduction
- Polynomial Rings and Polynomials
- Creation of Polynomial Rings
PolynomialRing(R, n): Rng, RngIntElt → RngMPolPolynomialAlgebra(R, n): Rng, RngIntElt → RngMPolPolynomialRing(R, n, order): Rng, RngIntElt, MonStgElt, ... → RngMPolPolynomialAlgebra(R, n, order): Rng, RngIntElt, MonStgElt, ... → RngMPol- Example: Assigning Generator Names
- Example: Global and Non-Global Polynomial Rings
- Print Names
- Graded Polynomial Rings
- Creation of Polynomials
P . i: RngMPol, RngInt → RngMPolEltelt< R | a >: RngMPol, RngElt → RngMPolEltR ! s: RngMPol, RngElt → RngMPolEltR ! s: RngMPol, [ RngElt ] → RngMPolEltelt< R | s >: RngMPol, [ RngElt ] → RngMPolEltMultivariatePolynomial(P, f, i): RngMPol, RngUPolElt, RngIntElt → RngMPolEltMultivariatePolynomial(P, f, v): RngMPol, RngUPolElt, RngMPolElt → RngMPolEltOne(P): RngMPol → RngMPolEltIdentity(P): RngMPol → RngMPolEltZero(P): RngMPol → RngMPolEltRepresentative(P): RngMPol → RngMPolElt
- Structure Operations
- Element Operations
- Arithmetic Operators
+ a: RngMPolElt → RngMPolElt- a: RngMPolElt → RngMPolElta + b: RngMPolElt, RngMPolElt → RngMPolElta - b: RngMPolElt, RngMPolElt → RngMPolElta * b: RngMPolElt, RngMPolElt → RngMPolElta ^ k: RngMPolElt, RngIntElt → RngMPolElta / b: RngMPolElt, RngMPolElt → FldFunMElta div b: RngMPolElt, RngMPolElt → RngMPolElta +:= b: RngMPolElt, RngMPolElt → RngMPolElta -:= b: RngMPolElt, RngMPolElt → RngMPolElta *:= b: RngMPolElt, RngMPolElt → RngMPolElta div:= b: RngMPolElt, RngMPolElt → RngMPolElt
- Equality and Membership
- Predicates on Ring Elements
- Coefficients, Monomials, and Terms
Coefficients(f): RngMPolElt → [ RngElt ]Coefficients(f, i): RngMPolElt, RngIntElt → [ RngElt ]Coefficients(f, v): RngMPolElt, RngMPolElt → [ RngElt ]Coefficient(f, i, k): RngMPolElt, RngIntElt, RngIntElt → RngEltCoefficient(f, v, k): RngMPolElt, RngMPolElt, RngIntElt → RngEltLeadingCoefficient(f): RngMPolElt → RngEltLeadingCoefficient(f, i): RngMPolElt, RngIntElt → RngEltLeadingCoefficient(f, v): RngMPolElt, RngMPolElt → RngEltLength(f): RngMPolElt → RngIntEltTrailingCoefficient(f): RngMPolElt → RngEltTrailingCoefficient(f, i): RngMPolElt, RngIntElt → RngEltTrailingCoefficient(f, v): RngMPolElt, RngMPolElt → RngEltCoefficientDenominator(f): RngMPolElt → RngEltCoefficientNumerator(f): RngMPolElt → RngEltMonomialCoefficient(f, m): RngMPolElt, RngMPolElt → RngEltMonomials(f): RngMPolElt → [ RngMPolElt ]CoefficientsAndMonomials(f): RngMPolElt → [ RngElt ], [ RngMPolElt ]LeadingMonomial(f): RngMPolElt → RngMPolEltTerms(f): RngMPolElt → [ RngMPolElt ]Terms(f, i): RngMPolElt, RngIntElt → [ RngMPolElt ]Terms(f, v): RngMPolElt, RngMPolElt → [ RngMPolElt ]Term(f, i, k): RngMPolElt, RngIntElt, RngIntElt → RngMPolEltTerm(f, v, k): RngMPolElt, RngMPolElt, RngIntElt → RngMPolEltLeadingTerm(f): RngMPolElt → RngMPolEltLeadingTerm(f, i): RngMPolElt, RngIntElt → RngMPolEltLeadingTerm(f, v): RngMPolElt, RngMPolElt → RngMPolEltTrailingTerm(f): RngMPolElt → RngEltTrailingTerm(f, i): RngMPolElt, RngIntElt → RngEltTrailingTerm(f, v): RngMPolElt, RngMPolElt → RngEltExponents(f): RngMPolElt → [ RngIntElt ]Monomial(P, E): RngMPol, [ RngIntElt ] → RngMPolEltPolynomial(C, M): [RngElt], [RngMPolElt] → RngMPolElt- Example: Coefficients and Monomials
- Degrees
- Univariate Polynomials
- Derivative, Integral
Derivative(f, i): RngMPolElt, RngIntElt → RngMPolEltDerivative(f, v): RngMPolElt, RngMPolElt → RngMPolEltDerivative(f, k, i): RngMPolElt, RngIntElt → RngMPolEltDerivative(f, k, v): RngMPolElt, RngMPolElt → RngMPolEltIntegral(f, i): RngMPolElt, RngIntElt → RngMPolEltIntegral(f, v): RngMPolElt, RngMPolElt → RngMPolEltJacobianMatrix( [ f ] ): [ RngMPolElt ] → RngMPol
- Evaluation, Interpolation
Evaluate(f, s): RngMPolElt, [ RngElt ] → RngEltEvaluate(f, s): RngMPolElt, < RngElt, ..., RngElt > → RngEltEvaluate(f, i, r): RngMPolElt, RngMPolElt, RngElt → RngMPolEltEvaluate(f, v, r): RngMPolElt, RngMPolElt, RngElt → RngMPolEltInterpolation(I, V, i): [ RngElt ], [ RngMPolElt ], RngIntElt → RngMPolEltInterpolation(I, V, v): [ RngElt ], [ RngMPolElt ], RngIntElt → RngMPolElt- Example: Polynomial Interpolation
- Quotient and Reductum
- Diagonalizing a Polynomial of Degree 2
- Greatest Common Divisors
- Common Divisors and Common Multiples
GreatestCommonDivisor(f, g): RngMPolElt, RngMPolElt → RngMPolEltGcd(f, g): RngMPolElt, RngMPolElt → RngMPolEltGCD(f, g): RngMPolElt, RngMPolElt → RngMPolEltGCD(Q): [ RngMPolElt ] → RngMPolEltLeastCommonMultiple(f, g): RngMPolElt, RngMPolElt → RngMPolEltLcm(f, g): RngMPolElt, RngMPolElt → RngMPolEltLCM(f, g): RngMPolElt, RngMPolElt → RngMPolEltLCM(Q): [ RngMPolElt ] → RngMPolEltNormalize(f): RngMPolElt → RngMPolEltClearDenominators(f): RngMPolElt → RngMPolEltClearDenominators(Q): [ RngMPolElt ] → [ RngMPolElt ]
- Content and Primitive Part
- Factorization and Irreducibility
- Resultants and Discriminants
Resultant(f, g, i): RngMPolElt, RngMPolElt, RngIntElt → RngMPolEltResultant(f, g, v): RngMPolElt, RngMPolElt, RngMPolElt → RngMPolEltDiscriminant(f, i): RngMPolElt, RngIntElt → RngMPolEltDiscriminant(f, v): RngMPolElt, RngMPolElt → RngMPolElt
- Polynomials over the Integers
- Bibliography