PolynomialFactorizationExplicit

catdef.spad line 1228 [edit on github]

This is the category of domains that know "enough" about themselves in order to factor univariate polynomials over their fraction field.

* : (%, %) -> %

from Magma

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associates? : (%, %) -> Boolean

from EntireRing

associator : (%, %, %) -> %

from NonAssociativeRng

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

charthRoot : % -> Union(%, "failed") if % has CharacteristicNonZero

charthRoot(r) returns the p-th root of r, or "failed" if none exists in the domain.

coerce : % -> %

from Algebra(%)

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

conditionP : Matrix(%) -> Union(Vector(%), "failed") if % has CharacteristicNonZero

conditionP(m) returns a vector of elements, not all zero, whose p-th powers (p is the characteristic of the domain) are a solution of the homogeneous linear system represented by m, or "failed" is there is no such vector.

exquo : (%, %) -> Union(%, "failed")

from EntireRing

factor : % -> Factored(%)

from UniqueFactorizationDomain

factorPolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%))

factorPolynomial(p) returns the factorization into irreducibles of the univariate polynomial p.

factorSquareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%))

factorSquareFreePolynomial(p) factors the univariate polynomial p into irreducibles where p is known to be square free and primitive with respect to its main variable.

gcd : (%, %) -> %

from GcdDomain

gcd : List(%) -> %

from GcdDomain

gcdPolynomial : (SparseUnivariatePolynomial(%), SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%)

gcdPolynomial(p, q) returns the gcd of the univariate polynomials p qnd q.

latex : % -> String

from SetCategory

lcm : (%, %) -> %

from GcdDomain

lcm : List(%) -> %

from GcdDomain

lcmCoef : (%, %) -> Record(llcm_res : %, coeff1 : %, coeff2 : %)

from LeftOreRing

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

leftRecip : % -> Union(%, "failed")

from MagmaWithUnit

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

prime? : % -> Boolean

from UniqueFactorizationDomain

recip : % -> Union(%, "failed")

from MagmaWithUnit

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

sample : () -> %

from AbelianMonoid

solveLinearPolynomialEquation : (List(SparseUnivariatePolynomial(%)), SparseUnivariatePolynomial(%)) -> Union(List(SparseUnivariatePolynomial(%)), "failed")

solveLinearPolynomialEquation([f1, ..., fn], g) (where the fi are relatively prime to each other) returns a list of ai such that g/prod fi = sum ai/fi or returns "failed" if no such list of ai's exists.

squareFree : % -> Factored(%)

from UniqueFactorizationDomain

squareFreePart : % -> %

from UniqueFactorizationDomain

squareFreePolynomial : SparseUnivariatePolynomial(%) -> Factored(SparseUnivariatePolynomial(%))

squareFreePolynomial(p) returns the square-free factorization of the univariate polynomial p.

subtractIfCan : (%, %) -> Union(%, "failed")

from CancellationAbelianMonoid

unit? : % -> Boolean

from EntireRing

unitCanonical : % -> %

from EntireRing

unitNormal : % -> Record(unit : %, canonical : %, associate : %)

from EntireRing

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

IntegralDomain

noZeroDivisors

Algebra(%)

RightModule(%)

Monoid

GcdDomain

EntireRing

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

AbelianGroup

LeftModule(%)

CommutativeStar

Module(%)

SetCategory

LeftOreRing

Rng

CommutativeRing

TwoSidedRecip

Magma

UniqueFactorizationDomain

SemiGroup

BiModule(%, %)

CoercibleTo(OutputForm)

AbelianSemiGroup

NonAssociativeSemiRing

NonAssociativeAlgebra(%)

NonAssociativeRng

unitsKnown

Ring

SemiRng

AbelianMonoid

NonAssociativeSemiRng

BasicType

SemiRing