XPolynomial(R)

xpoly.spad line 797 [edit on github]

• R : Ring

This type supports multivariate polynomials whose set of variables is Symbol. The representation is recursive. The coefficient ring may be non-commutative and the variables do not commute. However, coefficients and variables commute.

* : (%, %) -> %

from Magma

* : (%, R) -> %

from XFreeAlgebra(Symbol, R)

* : (R, %) -> %

from LeftModule(R)

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

* : (Symbol, %) -> %

from XFreeAlgebra(Symbol, R)

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

RemainderList : % -> List(Record(k : Symbol, c : %))

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

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

from NonAssociativeRng

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coef : (%, %) -> R

from XFreeAlgebra(Symbol, R)

coef : (%, FreeMonoid(Symbol)) -> R

from XFreeAlgebra(Symbol, R)

coefficient : (%, FreeMonoid(Symbol)) -> R

from FreeModuleCategory(R, FreeMonoid(Symbol))

coefficients : % -> List(R)

from FreeModuleCategory(R, FreeMonoid(Symbol))

coerce : R -> %

from XAlgebra(R)

coerce : FreeMonoid(Symbol) -> %

from CoercibleFrom(FreeMonoid(Symbol))

coerce : Integer -> %

from NonAssociativeRing

coerce : Symbol -> %

from XFreeAlgebra(Symbol, R)

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

constant : % -> R

from XFreeAlgebra(Symbol, R)

constant? : % -> Boolean

from XFreeAlgebra(Symbol, R)

construct : List(Record(k : FreeMonoid(Symbol), c : R)) -> %

from IndexedProductCategory(R, FreeMonoid(Symbol))

constructOrdered : List(Record(k : FreeMonoid(Symbol), c : R)) -> %

from IndexedProductCategory(R, FreeMonoid(Symbol))

degree : % -> NonNegativeInteger

from XPolynomialsCat(Symbol, R)

expand : % -> XDistributedPolynomial(Symbol, R)

latex : % -> String

from SetCategory

leadingCoefficient : % -> R

from IndexedProductCategory(R, FreeMonoid(Symbol))

leadingMonomial : % -> %

from IndexedProductCategory(R, FreeMonoid(Symbol))

leadingSupport : % -> FreeMonoid(Symbol)

from IndexedProductCategory(R, FreeMonoid(Symbol))

leadingTerm : % -> Record(k : FreeMonoid(Symbol), c : R)

from IndexedProductCategory(R, FreeMonoid(Symbol))

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

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

from MagmaWithUnit

linearExtend : (Mapping(R, FreeMonoid(Symbol)), %) -> R if R has CommutativeRing

from FreeModuleCategory(R, FreeMonoid(Symbol))

listOfTerms : % -> List(Record(k : FreeMonoid(Symbol), c : R))

from IndexedDirectProductCategory(R, FreeMonoid(Symbol))

lquo : (%, %) -> %

from XFreeAlgebra(Symbol, R)

lquo : (%, FreeMonoid(Symbol)) -> %

from XFreeAlgebra(Symbol, R)

lquo : (%, Symbol) -> %

from XFreeAlgebra(Symbol, R)

map : (Mapping(R, R), %) -> %

from XFreeAlgebra(Symbol, R)

maxdeg : % -> FreeMonoid(Symbol)

from XPolynomialsCat(Symbol, R)

mindeg : % -> FreeMonoid(Symbol)

from XFreeAlgebra(Symbol, R)

mindegTerm : % -> Record(k : FreeMonoid(Symbol), c : R)

from XFreeAlgebra(Symbol, R)

mirror : % -> %

from XFreeAlgebra(Symbol, R)

monomial : (R, FreeMonoid(Symbol)) -> %

from XFreeAlgebra(Symbol, R)

monomial? : % -> Boolean

from XFreeAlgebra(Symbol, R)

monomials : % -> List(%)

from FreeModuleCategory(R, FreeMonoid(Symbol))

numberOfMonomials : % -> NonNegativeInteger

from IndexedDirectProductCategory(R, FreeMonoid(Symbol))

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

plenaryPower : (%, PositiveInteger) -> % if R has CommutativeRing

from NonAssociativeAlgebra(R)

quasiRegular : % -> %

from XFreeAlgebra(Symbol, R)

quasiRegular? : % -> Boolean

from XFreeAlgebra(Symbol, R)

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

from MagmaWithUnit

reductum : % -> %

from IndexedProductCategory(R, FreeMonoid(Symbol))

retract : % -> R

from RetractableTo(R)

retract : % -> FreeMonoid(Symbol)

from RetractableTo(FreeMonoid(Symbol))

retractIfCan : % -> Union(R, "failed")

from RetractableTo(R)

retractIfCan : % -> Union(FreeMonoid(Symbol), "failed")

from RetractableTo(FreeMonoid(Symbol))

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

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

from MagmaWithUnit

rquo : (%, %) -> %

from XFreeAlgebra(Symbol, R)

rquo : (%, FreeMonoid(Symbol)) -> %

from XFreeAlgebra(Symbol, R)

rquo : (%, Symbol) -> %

from XFreeAlgebra(Symbol, R)

sample : () -> %

from AbelianMonoid

sh : (%, %) -> % if R has CommutativeRing

from XFreeAlgebra(Symbol, R)

sh : (%, NonNegativeInteger) -> % if R has CommutativeRing

from XFreeAlgebra(Symbol, R)

smaller? : (%, %) -> Boolean if R has Comparable

from Comparable

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

from CancellationAbelianMonoid

support : % -> List(FreeMonoid(Symbol))

from FreeModuleCategory(R, FreeMonoid(Symbol))

trunc : (%, NonNegativeInteger) -> %

from XPolynomialsCat(Symbol, R)

unexpand : XDistributedPolynomial(Symbol, R) -> %

varList : % -> List(Symbol)

from XFreeAlgebra(Symbol, R)

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

Comparable

noZeroDivisors

CoercibleFrom(R)

Algebra(R)

RightModule(%)

Monoid

XPolynomialsCat(Symbol, R)

BiModule(R, R)

NonAssociativeAlgebra(R)

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

RightModule(R)

LeftModule(%)

LeftModule(R)

CoercibleFrom(FreeMonoid(Symbol))

SetCategory

CoercibleTo(OutputForm)

Rng

SemiGroup

Magma

XAlgebra(R)

Ring

RetractableTo(FreeMonoid(Symbol))

BiModule(%, %)

unitsKnown

AbelianGroup

AbelianSemiGroup

FreeModuleCategory(R, FreeMonoid(Symbol))

XFreeAlgebra(Symbol, R)

NonAssociativeSemiRing

AbelianProductCategory(R)

IndexedProductCategory(R, FreeMonoid(Symbol))

Module(R)

IndexedDirectProductCategory(R, FreeMonoid(Symbol))

RetractableTo(R)

NonAssociativeRng

SemiRng

AbelianMonoid

NonAssociativeSemiRng

BasicType

SemiRing