Univariate Polynomial Rings
- Introduction
- Creation Functions
- Creation of Structures
- Print Options
- Creation of Elements
P . 1: RngUPol, RngInt → RngPolEltelt< P | a₀, ..., a_d >: RngUPol, RngElt, ..., RngElt → RngUPolEltP ! s: RngUPol, RngElt → RngPolEltP ! s: RngUPol, [ RngElt ] → RngPolEltelt< P | s >: RngUPol, [ RngElt ] → RngUPolEltPolynomial(Q): [ RngElt ] → RngUPolEltPolynomial(R, Q): Rng, [ RngElt] → RngUPolEltPolynomial(R, f): Rng, RngUPolElt → RngUPolEltOne(P): RngUPol → RngUPolEltIdentity(P): RngUPol → RngUPolEltZero(P): RngUPol → RngUPolEltRepresentative(P): RngUPol → RngUPolElt- Example: Creating Polynomials
- Structure Operations
- Element Operations
- Parent and Category
- Arithmetic Operators
+ a: RngUPolElt → RngUPolElt- a: RngUPolElt → RngUPolElta + b: RngUPolElt, RngUPolElt → RngUPolElta - b: RngUPolElt, RngUPolElt → RngUPolElta * b: RngUPolElt, RngUPolElt → RngUPolElta ^ k: RngUPolElt, RngIntElt → RngUPolElta / b: RngUPolElt, RngElt → FldFunUElta div b: RngUPolElt, RngUPolElt → RngUPolElta mod b: RngUPolElt, RngUPolElt → RngUPolElta +:= b: RngUPolElt, RngUPolElt → RngUPolElta -:= b: RngUPolElt, RngUPolElt → RngUPolElta *:= b: RngUPolElt, RngUPolElt → RngUPolElt
- Equality and Membership
- Predicates on Ring Elements
- Coefficients and Terms
- Degree
- Roots
Roots(p): RngUPolElt → [ < RngElt, RngIntElt> ]Roots(p, S): RngUPolElt → [ < RngElt, RngIntElt> ]HasRoot(p): RngUPolElt → BoolElt, RngEltHasRoot(p, S): RngUPolElt, Rng → BoolElt, RngEltSmallRoots(p, N, X): RngUPolElt, RngElt, RngElt → [RngElt]- Example: Factoring an RSA Modulus with Small Roots
SetVerbose("SmallRoots", v): MonStgElt, RngIntElt
- Derivative, Integral
- Evaluation, Interpolation
- Decomposition
- Quotient and Remainder
Quotrem(f, g): RngUPolElt, RngUPolElt → RngUPolElt, RngUPolEltf div g: RngUPolElt, RngUPolElt → RngUPolEltIsDivisibleBy(f, g): RngUPolElt, RngUPolElt → BoolElt, RngUPolEltExactQuotient(f, g): RngUPolElt, RngUPolElt → RngUPolEltf mod g: RngUPolElt, RngUPolElt → RngUPolEltValuation(f, g): RngUPolElt, RngUPolElt → RngIntEltReductum(f): RngUPolElt → RngUPolEltPseudoRemainder(f, g): RngUPolElt, RngUPolElt → RngUPolEltEuclideanNorm(p): RngUPol → RngIntElt
- Modular Arithmetic
- Other Operations
- Common Divisors and Common Multiples
- Common Divisors and Common Multiples
GreatestCommonDivisor(f, g): RngUPolElt, RngUPolElt → RngUPolEltGcd(f, g): RngUPolElt, RngUPolElt → RngUPolEltGCD(f, g): RngUPolElt, RngUPolElt → RngUPolEltExtendedGreatestCommonDivisor(f, g): RngUPolElt, RngUPolElt → RngUPolElt, RngUPolElt, RngUPolEltXgcd(f, g): RngUPolElt, RngUPolElt → RngUPolElt, RngUPolElt, RngUPolEltXGCD(f, g): RngUPolElt, RngUPolElt → RngUPolElt, RngUPolElt, RngUPolEltLeastCommonMultiple(f, g): RngUPolElt, RngUPolElt → RngUPolEltLcm(f, g): RngUPolElt, RngUPolElt → RngUPolEltLCM(f, g): RngUPolElt, RngUPolElt → RngUPolEltNormalize(f): RngUPolElt → RngUPolElt
- Content and Primitive Part
- Polynomials over the Integers
- Polynomials over Finite Fields
PrimePolynomials(R, d): RngUPol, RngIntElt → SeqEnum[ RngUPolElt ]PrimePolynomials(R, d, n): RngUPol, RngIntElt, RngIntElt → SeqEnum[ RngUPolElt ]RandomPrimePolynomial(R, d): RngUPol, RngIntElt → RngUPolEltNumberOfPrimePolynomials(q, d): RngIntElt, RngIntElt → RngIntEltNumberOfPrimePolynomials(K, d): FldFin, RngIntElt → RngIntEltNumberOfPrimePolynomials(R, d): RngUPol, RngIntElt → RngIntEltJacobiSymbol(a,b): RngUPol, RngUPol → RngIntElt
- Factorization
- Factorization and Irreducibility
Factorization(f): RngUPolElt → [ < RngUPolElt, RngIntElt >], RngEltFactorisation(f): RngUPolElt → [ < RngUPolElt, RngIntElt >], RngEltHasPolynomialFactorization(R): Rng → BoolEltSetVerbose("PolyFact", v): MonStgElt, RngIntEltFactorisationToPolynomial(f): [Tup] → BoolEltFacpol(f): [Tup] → BoolElt- Example: Factoring Swinnerton-Dyer Polynomials
SquarefreeFactorization(f): RngUPolElt → [ <RngUPolElt, RngIntElt> ]DistinctDegreeFactorization(f): RngUPolElt → [ <RngIntElt, RngUPolElt> ]EqualDegreeFactorization(f, d, g): RngUPolElt, RngIntElt, RngUPolElt → [ RngUPolElt ]IsIrreducible(f): RngUPolElt → BoolEltIsSeparable(f): RngUPolElt → BoolEltQMatrix(f): RngUPolElt → AlgMatElt
- Sylvester Matrix, Resultant, and Discriminant
- Hensel Lifting
- Ideals and Quotient Rings
- Creation of Ideals and Quotients
ideal< R | a₁, ..., aᵣ >: RngUPol, RngUPolElt, ..., RngUPolElt → RngUPolquo< R | I >: RngUPol, RngUPol → RngUPolResquo< R | a₁, ..., aᵣ >: RngUPol, RngUPolElt, ..., RngUPolElt → RngUPolRes, Map
- Ideal Arithmetic
- Other Functions on Ideals
- Other Functions on Quotients
- Special Families of Polynomials
- Bibliography