Polynomial Rings and Polynomials
Creation of Polynomial Rings
PolynomialRing(R, n) : Rng, RngIntElt -> RngMPol
PolynomialRing(R, n, order) : Rng, RngIntElt, MonStgElt, ... -> RngMPol
Example RngMPol_AssignNames (H24E1)
Example RngMPol_Global (H24E2)
Print Names
AssignNames(~P, s) : RngMPol, [ MonStgElt ]) ->
Name(P, i) : RngMPol, RngIntElt -> RngMPolElt
Creation of Polynomials
P . i : RngMPol, RngInt -> RngMPolElt
elt< R | a > : RngMPol, RngElt -> RngMPolElt
MultivariatePolynomial(P, f, i) : RngMPol, RngUPolElt, RngIntElt -> RngMPolElt
Related Structures
BaseRing(P) : RngMPol -> Rng
Numerical Invariants
Rank(P) : RngMPol -> RngIntElt
Changing Coefficient Ring
ChangeRing(P, S) : RngMPol, Rng -> RngMPol
Homomorphisms
hom< P -> S | f, y1, ..., yn > : RngMPol, Rng -> Map
Example RngMPol_Homomorphism (H24E3)
Predicates on Ring Elements
IsDivisibleBy(a, b) : RngMPolElt, RngMPolElt -> BoolElt, RngMPolElt
IsAlgebraicallyDependent(S) : RngMPolElt -> BoolElt
Coefficients, Monomials and Terms
Coefficients(f) : RngMPolElt -> [ RngElt ]
Coefficients(f, i) : RngMPolElt, RngIntElt -> [ RngElt ]
Coefficient(f, i, k) : RngMPolElt, RngIntElt, RngIntElt -> RngElt
LeadingCoefficient(f) : RngMPolElt -> RngElt
LeadingCoefficient(f, i) : RngMPolElt, RngIntElt -> RngElt
Length(f) : RngMPolElt -> RngIntElt
TrailingCoefficient(f) : RngMPolElt -> RngElt
TrailingCoefficient(f, i) : RngMPolElt, RngIntElt -> RngElt
MonomialCoefficient(f, m) : RngMPolElt, RngMPolElt -> RngElt
Monomials(f) : RngMPolElt -> [ RngMPolElt ]
CoefficientsAndMonomials(f) : RngMPolElt -> [ RngElt ], [ RngMPolElt ]
LeadingMonomial(f) : RngMPolElt -> RngMPolElt
Terms(f) : RngMPolElt -> [ RngMPolElt ]
Terms(f, i) : RngMPolElt, RngIntElt -> [ RngMPolElt ]
Term(f, i, k) : RngMPolElt, RngIntElt, RngIntElt -> RngMPolElt
LeadingTerm(f) : RngMPolElt -> RngMPolElt
LeadingTerm(f, i) : RngMPolElt, RngIntElt -> RngMPolElt
TrailingTerm(f) : RngMPolElt -> RngElt
TrailingTerm(f, i) : RngMPolElt, RngIntElt -> RngElt
Exponents(f) : RngMPolElt -> [ RngIntElt ]
Monomial(P, E) : RngMPol, [ RngIntElt ] -> RngMPolElt
Polynomial(C, M) : [RngElt], [RngMPolElt] -> RngMPolElt
Example RngMPol_Coefficients (H24E4)
Degrees
Degree(f, i) : RngMPolElt, RngIntElt -> RngIntElt
TotalDegree(f) : RngMPolElt -> RngIntElt
LeadingTotalDegree(f) : RngMPolElt -> RngIntElt
Univariate Polynomials
IsUnivariate(f) : RngMPolElt -> BoolElt, RngUPolElt, RngIntElt
IsUnivariate(f, i) : RngMPolElt, RngIntElt -> BoolElt, RngUPolElt
UnivariatePolynomial(f) : RngMPolElt -> RngUPolElt
Example RngMPol_UnivariatePolynomial (H24E5)
Derivative, Integral
Derivative(f, i) : RngMPolElt, RngIntElt -> RngMPolElt
Derivative(f, k, i) : RngMPolElt, RngIntElt -> RngMPolElt
Integral(f, i) : RngMPolElt, RngIntElt -> RngMPolElt
JacobianMatrix( [ f ] ) : [ RngMPolElt ] -> RngMPol
Evaluation, Interpolation
Evaluate(f, s) : RngMPolElt, [ RngElt ] -> RngElt
Evaluate(f, i, r) : RngMPolElt, RngMPolElt, RngElt -> RngMPolElt
Interpolation(I, V, i) : [ RngElt ], [ RngMPolElt ], RngIntElt -> RngMPolElt
Example RngMPol_Interpolate (H24E6)
Quotient and Reductum
f div g : RngMPolElt, RngMPolElt -> RngMPolElt
Reductum(f) : RngMPolElt -> RngMPolElt
Reductum(f, i) : RngMPolElt, RngIntElt -> RngMPolElt
Diagonalizing a Polynomial of Degree 2
SymmetricBilinearForm(f) : RngMPolElt -> ModMatRngElt
DiagonalForm(f) : RngMPolElt -> RngMPolElt, ModMatRngElt
Example RngMPol_Sym_Bi_Linear (H24E7)
Common Divisors and Common Multiples
GreatestCommonDivisor(f, g) : RngMPolElt, RngMPolElt -> RngMPolElt
GCD(Q) : [ RngMPolElt ] -> RngMPolElt
LeastCommonMultiple(f, g) : RngMPolElt, RngMPolElt -> RngMPolElt
LCM(Q) : [ RngMPolElt ] -> RngMPolElt
Normalize(f) : RngMPolElt -> RngMPolElt
ClearDenominators(f) : RngMPolElt -> RngMPolElt
Content and Primitive Part
Content(f) : RngMPolElt -> RngIntElt
PrimitivePart(f) : RngMPolElt -> RngMPolElt
ContentAndPrimitivePart(f) : RngMPolElt -> RngIntElt, RngMPolElt
Factorization and Irreducibility
Factorization(f) : RngMPolElt -> [ < RngMPolElt, RngIntElt >], RngElt
SquarefreeFactorization(f) : RngMPolElt -> [ <RngMPolElt, RngIntElt> ]
SquarefreePart(f) : RngMPolElt -> RngMPolElt
IsIrreducible(f) : RngMPolElt -> BoolElt
SetVerbose("PolyFact", v) : MonStgElt, RngIntElt ->
Example RngMPol_Trinomials (H24E8)
Example RngMPol_Vandermonde (H24E9)
Example RngMPol_Heron (H24E10)
Example RngMPol_FiniteFieldFactorization (H24E11)
Resultants and Discriminants
Resultant(f, g, i) : RngMPolElt, RngMPolElt, RngIntElt -> RngMPolElt
Discriminant(f, i) : RngMPolElt, RngIntElt -> RngMPolElt
Polynomials over the Integers
Sign(f) : RngMPolElt -> RngIntElt
AbsoluteValue(f) : RngMPolElt -> RngMPolElt
MaxNorm(f) : RngMPolElt -> RngIntElt
SumNorm(f) : RngMPolElt -> RngIntElt
Bibliography
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Version: V2.19 of
Wed Apr 24 15:09:57 EST 2013