UNIVARIATE POLYNOMIAL RINGS
Acknowledgements
Introduction
Representation
Creation Functions
Creation of Structures
Print Options
Creation of Elements
Structure Operations
Related Structures
Changing Rings
Numerical Invariants
Ring Predicates and Booleans
Homomorphisms
Element Operations
Parent and Category
Arithmetic Operators
Equality and Membership
Predicates on Ring Elements
Coefficients and Terms
Degree
Roots
Derivative, Integral
Evaluation, Interpolation
Decomposition
Quotient and Remainder
Modular Arithmetic
Other Operations
Common Divisors and Common Multiples
Common Divisors and Common Multiples
Content and Primitive Part
Polynomials over the Integers
Polynomials over Finite Fields
Factorization
Factorization and Irreducibility
Sylvester Matrix, Resultant and Discriminant
Hensel Lifting
Ideals and Quotient Rings
Creation of Ideals and Quotients
Ideal Arithmetic
Other Functions on Ideals
Other Functions on Quotients
Special Families of Polynomials
Orthogonal Polynomials
Permutation Polynomials
The Bernoulli Polynomial
Swinnerton-Dyer Polynomials
Bibliography
Introduction
Representation
Creation Functions
Creation of Structures
PolynomialAlgebra(R) : Rng -> RngUPol
Example RngPol_Creation (H24E1)
Print Options
AssignNames(~P, s) : RngUPol, [ MonStgElt ]) ->
Name(P, i) : RngUPol, RngIntElt -> RngUPolElt
Creation of Elements
P . 1 : RngUPol, RngInt -> RngPolElt
elt< P | a0, ..., ad > : 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 (H24E2)
Structure Operations
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 (H24E3)
Numerical Invariants
Rank(P) : RngUPol -> RngIntElt
# P : RngUPolRes -> RngIntElt
Ring Predicates and Booleans
Homomorphisms
hom< P -> S | f, y > : RngUPol, Rng, Map, RngElt -> Map
Example RngPol_Homomorphism (H24E4)
Element Operations
Parent and Category
Arithmetic Operators
Equality and Membership
Predicates on Ring Elements
Coefficients and Terms
Coefficients(p) : RngUPolElt -> [ RngElt ]
Coefficient(p, i) : RngUPolElt, RngIntElt -> RngElt
MonomialCoefficient(p, m) : RngUPolElt, RngUPolElt -> RngElt
LeadingCoefficient(p) : RngUPolElt -> RngElt
TrailingCoefficient(p) : RngUPolElt -> RngElt
ConstantCoefficient(p) : RngUPolElt -> RngElt
Terms(p) : RngUPolElt -> [ RngUPolElt ]
LeadingTerm(p) : RngUPolElt -> RngUPolElt
TrailingTerm(p) : RngUPolElt -> RngUPolElt
Monomials(p) : RngUPolElt -> SeqEnum
Support(p) : RngUPolElt -> [RngIntElt], [RngElt]
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
SmallRoots(p, N, X) : RngUPolElt, RngElt, RngElt -> [RngElt]
Example RngPol_SmallRootsUsage (H24E5)
SetVerbose("SmallRoots", v) : MonStgElt, RngIntElt ->
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
Decomposition
Decomposition(f) : RngUPolElt -> [[RngUPolElt]]
Example RngPol_decomp-ex (H24E6)
Quotient and Remainder
Quotrem(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt, RngUPolElt
f div g : RngUPolElt, RngUPolElt -> RngUPolElt
IsDivisibleBy(f, g) : RngUPolElt, RngUPolElt -> BoolElt, RngUPolElt
ExactQuotient(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
f mod g : RngUPolElt, RngUPolElt -> RngUPolElt
Valuation(f, g) : RngUPolElt, RngUPolElt -> RngIntElt
Reductum(f) : RngUPolElt -> RngUPolElt
PseudoRemainder(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
EuclideanNorm(p) : RngUPol -> RngIntElt
Modular Arithmetic
Modexp(f, n, g) : RngUPolElt, RngIntElt, RngUPolElt -> RngUPolElt
ChineseRemainderTheorem(X, M) : [RngUPolElt], [RngUPolElt] -> RngUPolElt
Other Operations
ReciprocalPolynomial(f) : RngUPolElt -> RngUPolElt
PowerPolynomial(f,n) : RngUPolElt, RngIntElt -> RngUPolElt
f ^ M : RngUPolElt, Mtrx -> 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
JacobiSymbol(a,b) : RngUPol, RngUPol -> RngIntElt
Factorization
Factorization and Irreducibility
Factorization(f) : RngUPolElt -> [ < RngUPolElt, RngIntElt >], RngElt
HasPolynomialFactorization(R) : Rng -> BoolElt
SetVerbose("PolyFact", v) : MonStgElt, RngIntElt ->
FactorisationToPolynomial(f) : [Tup] -> BoolElt
Example RngPol_SwinnertonDyerPolynomial (H24E7)
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
Sylvester Matrix, Resultant and Discriminant
SylvesterMatrix(f, g) : RngUPolElt, RngUPolElt -> Mtrx
Resultant(f, g) : RngUPolElt, RngUPolElt -> RngElt
Discriminant(f) : RngUPolElt -> RngIntElt
CompanionMatrix(f) : RngUPolElt -> AlgMatElt
Hensel Lifting
HenselLift(f, s, P) : RngUPolElt, [ RngUPolElt ], RngUPol -> [ RngUPolElt ]
Example RngPol_Hensel (H24E8)
Ideals and Quotient Rings
Creation of Ideals and Quotients
ideal< R | a1, ..., ar > : 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
Swinnerton-Dyer Polynomials
SwinnertonDyerPolynomial(n) : RngIntElt -> RngUPolElt
Bibliography
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