Creation of Structures
PolynomialAlgebra(R) : Rng -> RngUPol
Example RngPol_Creation (H44E1)
Print Options
AssignNames(~P, s) : RngUPol, [ MonStgElt ]) ->
Name(P, i) : RngUPol, RngIntElt -> RngPolElt
Creation of Elements
P . 1 : RngUPol, RngInt -> RngPolElt
elt< P | a_0, ..., a_d > : RngUPol, RngElt, ..., RngElt -> RngUPolElt
P ! s : RngUPol, RngElt -> RngPolElt
Polynomial(Q) : [ RngElt ] -> RngUPolElt
Polynomial(R, Q) : Rng, [ RngElt] -> RngUPolElt
Polynomial(R, f) : Rng, RngUPolElt -> RngUPolElt
Example RngPol_Polynomials (H44E2)
Related Structures
BaseRing(P) : RngUPol -> Rng
Changing Rings
ChangeRing(P, S) : RngUPol, Rng -> RngUPol, Map
ChangeRing(P, S, f) : RngUPol, Rng, Map -> RngUPol, Map
Example RngPol_ChangeRing (H44E3)
Numerical Invariants
Rank(P) : RngUPol -> RngIntElt
# P : RngUPolRes -> RngIntElt
Homomorphisms
hom< P -> S | f, y > : RngUPol, Rng, Map, RngElt -> Map
Example RngPol_Homomorphism (H44E4)
Coefficients and Terms
Coefficients(p) : RngUPolElt -> [ RngElt ]
Coefficient(p, i) : RngUPolElt, RngIntElt -> RngElt
MonomialCoefficient(p, m) : RngUPolElt, RngUPolElt -> RngElt
LeadingCoefficient(f) : RngUPolElt -> RngElt
TrailingCoefficient(p) : RngUPolElt -> RngElt
Terms(p) : RngUPolElt -> [ RngUPolElt ]
LeadingTerm(p) : RngUPolElt -> RngUPolElt
TrailingTerm(p) : RngUPolElt -> RngUPolElt
Round(p) : RngUPolElt -> RngUPolElt
Valuation(p) : RngUPolElt -> RngIntElt
Degree
Degree(p) : RngUPolElt -> RngIntElt
Roots
Roots(p) : RngUPolElt -> [ < RngElt, RngIntElt> ]
Roots(p, S) : RngUPolElt -> [ < RngElt, RngIntElt> ]
HasRoot(p) : RngUPolElt -> BoolElt, RngElt
HasRoot(p, S) : RngUPolElt, Rng -> BoolElt, RngElt
Derivative, Integral
Derivative(p) : RngUPolElt -> RngUPolElt
Derivative(p, n) : RngUPolElt, RngIntElt -> RngUPolElt
Integral(p) : RngUPolElt -> RngUPolElt
Evaluation, Interpolation
Evaluate(p, r) : RngUPolElt, RngElt -> RngElt
Interpolation(I, V) : [ RngElt ], [ RngElt ] -> RngUPolElt
Quotient and Remainder
Quotrem(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt, RngUPolElt
f div g : RngUPolElt, RngUPolElt -> RngUPolElt
ExactQuotient(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
f mod g : RngUPolElt, RngUPolElt -> RngUPolElt
Reductum(f) : RngUPolElt -> RngUPolElt
PseudoRemainder(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
EuclideanNorm(p) : RngUPol -> RngIntElt
Modular Arithmetic
Modexp(f, n, g) : RngUPolElt, RngIntElt, RngUPolElt -> RngUPolElt
Common Divisors and Common Multiples
Common Divisors and Common Multiples
GreatestCommonDivisor(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
ExtendedGreatestCommonDivisor(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt, RngUPolElt, RngUPolElt
LeastCommonMultiple(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
Normalize(f) : RngUPolElt -> RngUPolElt
Content and Primitive Part
Content(p) : RngUPolElt -> RngIntElt
PrimitivePart(p) : RngUPolElt -> RngUPolElt
ContentAndPrimitivePart(p) : RngUPolElt -> RngIntElt, RngUPolElt
Polynomials over the Integers
Sign(p) : RngUPolElt -> RngIntElt
AbsoluteValue(p) : RngUPolElt -> RngUPolElt
MaxNorm(p) : RngUPolElt -> RngIntElt
SumNorm(p) : RngUPolElt -> RngIntElt
DedekindTest(p, m) : RngUPolElt, RngIntElt -> Boolelt
Polynomials over finite fields
PrimePolynomials(R, d) : RngUPol, RngIntElt -> SeqEnum[ RngUPolElt ]
RandomPrimePolynomial(R, d) : RngUPol, RngIntElt -> RngUPolElt
NumberOfPrimePolynomials(q, d) : RngIntElt, RngIntElt -> RngIntElt
Factorization and Irreducibility
Factorization(f) : RngUPolElt -> [ < RngUPolElt, RngIntElt >], RngElt
HasPolynomialFactorization(R) : Rng -> BoolElt
SetVerbose("PolyFact", v) : MonStgElt, RngIntElt ->
Example RngPol_SwinnertonDyerPolynomial (H44E5)
SquarefreeFactorization(f) : RngUPolElt -> [ <RngUPolElt, RngIntElt> ]
DistinctDegreeFactorization(f) : RngUPolElt -> [ <RngIntElt, RngUPolElt> ]
EqualDegreeFactorization(f, d, g) : RngUPolElt, RngIntElt, RngUPolElt -> [ RngUPolElt ]
IsIrreducible(f) : RngUPolElt -> BoolElt
IsSeparable(f) : RngUPolElt -> BoolElt
QMatrix(f) : RngUPolElt -> AlgMatElt
Resultant and Discriminant
Discriminant(f) : RngUPolElt -> RngIntElt
Resultant(f, g) : RngUPolElt, RngUPolElt -> RngElt
CompanionMatrix(f) : RngUPolElt -> AlgMatElt
Hensel Lifting
HenselLift(f, s, P) : RngUPolElt, [ RngUPolElt ], RngRes -> [ RngUPolElt ]
Example RngPol_Hensel (H44E6)
Creation of Ideals and Quotients
ideal< R | a_1, ..., a_r > : RngUPol, RngUPolElt, ..., RngUPolElt -> RngUPol
quo< R | I > : RngUPol, RngUPol -> RngUPolRes
Ideal Arithmetic
I + J : RngUPol, RngUPol -> RngUPol
I * J : RngUPol, RngUPol -> RngUPol
I meet J : RngUPol, RngUPol -> RngUPol
a in I : RngUPolElt, RngUPol -> BoolElt
a notin I : RngUPolElt, RngUPol -> BoolElt
I eq J : RngUPol, RngUPol -> BoolElt
I ne J : RngUPol, RngUPol -> BoolElt
I subset J : RngUPol, RngUPol -> BoolElt
I notsubset J : RngUPol, RngUPol -> BoolElt
Other Functions on Ideals
I . 1 : RngUPol -> RngUPolElt
Other Functions on Quotients
Modulus(Q) : RngUPolRes -> RngUPolElt
PreimageRing(Q) : RngUPolRes -> RngUPol
Special Families of Polynomials
Orthogonal Polynomials
ChebyshevFirst(n) : RngIntElt -> RngUPolElt
ChebyshevSecond(n) : RngIntElt -> RngUPolElt
LegendrePolynomial(n) : RngIntElt -> RngUPolElt
LaguerrePolynomial(n) : RngIntElt -> RngUPolElt
HermitePolynomial(n) : RngIntElt -> RngUPolElt
GegenbauerPolynomial(n, m) : RngIntElt, RngElt ->RngUPolElt
Permutation Polynomials
DicksonFirst(n, a) : RngIntElt, RngElt -> RngUPolElt
DicksonSecond(n, a) : RngIntElt, RngElt -> RngUPolElt
The Bernoulli Polynomial
BernoulliPolynomial(n) : RngIntElt -> RngUPolElt