XPolynomialRing(R, E)ΒΆ

xpoly.spad line 134

This domain represents generalized polynomials with coefficients (from a not necessarily commutative ring), and words belonging to an arbitrary OrderedMonoid. This type is used, for instance, by the XDistributedPolynomial domain constructor where the Monoid is free.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
#: % -> NonNegativeInteger
\# p returns the number of terms in p.
*: (%, %) -> %
from Magma
*: (%, R) -> %
p*r returns the product of p by r.
*: (E, R) -> %
from FreeModuleCategory(R, E)
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> %
from LeftModule R
*: (R, E) -> %
from FreeModuleCategory(R, E)
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
/: (%, R) -> % if R has Field
p/r returns p*(1/r).
<: (%, %) -> Boolean if R has OrderedAbelianMonoid
from PartialOrder
<=: (%, %) -> Boolean if R has OrderedAbelianMonoid
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean if R has OrderedAbelianMonoid
from PartialOrder
>=: (%, %) -> Boolean if R has OrderedAbelianMonoid
from PartialOrder
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
associator: (%, %, %) -> %
from NonAssociativeRng
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
coef: (%, E) -> R
coef(p, e) extracts the coefficient of the monomial e. Returns zero if e is not present.
coefficient: (%, E) -> R
from FreeModuleCategory(R, E)
coefficients: % -> List R
from FreeModuleCategory(R, E)
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: E -> %
coerce(e) returns 1*e
coerce: Integer -> %
from NonAssociativeRing
coerce: R -> %
from XAlgebra R
commutator: (%, %) -> %
from NonAssociativeRng
constant: % -> R
constant(p) return the constant term of p.
constant?: % -> Boolean
constant?(p) tests whether the polynomial p belongs to the coefficient ring.
construct: List Record(k: E, c: R) -> %
from IndexedDirectProductCategory(R, E)
constructOrdered: List Record(k: E, c: R) -> %
from IndexedDirectProductCategory(R, E)
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
latex: % -> String
from SetCategory
leadingCoefficient: % -> R
from IndexedDirectProductCategory(R, E)
leadingMonomial: % -> %
from IndexedDirectProductCategory(R, E)
leadingSupport: % -> E
from IndexedDirectProductCategory(R, E)
leadingTerm: % -> Record(k: E, c: R)
from IndexedDirectProductCategory(R, E)
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
linearExtend: (E -> R, %) -> R if R has CommutativeRing
from FreeModuleCategory(R, E)
listOfTerms: % -> List Record(k: E, c: R)
from IndexedDirectProductCategory(R, E)
map: (R -> R, %) -> %
map(fn, x) returns Sum(fn(r_i) w_i) if x writes Sum(r_i w_i).
max: (%, %) -> % if R has OrderedAbelianMonoid
from OrderedSet
maxdeg: % -> E
maxdeg(p) returns the greatest word occurring in the polynomial p with a non-zero coefficient. An error is produced if p is zero.
min: (%, %) -> % if R has OrderedAbelianMonoid
from OrderedSet
mindeg: % -> E
mindeg(p) returns the smallest word occurring in the polynomial p with a non-zero coefficient. An error is produced if p is zero.
monomial: (R, E) -> %
from IndexedDirectProductCategory(R, E)
monomial?: % -> Boolean
from IndexedDirectProductCategory(R, E)
monomials: % -> List %
from FreeModuleCategory(R, E)
numberOfMonomials: % -> NonNegativeInteger
from IndexedDirectProductCategory(R, E)
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
quasiRegular: % -> %
quasiRegular(x) return x minus its constant term.
quasiRegular?: % -> Boolean
quasiRegular?(x) return true if constant(p) is zero.
recip: % -> Union(%, failed)
from MagmaWithUnit
reductum: % -> %
reductum(p) returns p minus its leading term. An error is produced if p is zero.
retract: % -> E
from RetractableTo E
retractIfCan: % -> Union(E, failed)
from RetractableTo E
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from AbelianMonoid
smaller?: (%, %) -> Boolean if R has Comparable
from Comparable
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
sup: (%, %) -> % if R has OrderedAbelianMonoidSup
from OrderedAbelianMonoidSup
support: % -> List E
from FreeModuleCategory(R, E)
totalDegree: % -> NonNegativeInteger if E has coerce: E -> Vector NonNegativeInteger
from FreeModuleCategory(R, E)
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianProductCategory R

AbelianSemiGroup

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid

canonicalUnitNormal if R has canonicalUnitNormal

CoercibleTo OutputForm

Comparable if R has Comparable

FreeModuleCategory(R, E)

IndexedDirectProductCategory(R, E)

LeftModule %

LeftModule R

Magma

MagmaWithUnit

Module R if R has CommutativeRing

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if R has noZeroDivisors

OrderedAbelianMonoid if R has OrderedAbelianMonoid

OrderedAbelianMonoidSup if R has OrderedAbelianMonoidSup

OrderedAbelianSemiGroup if R has OrderedAbelianMonoid

OrderedCancellationAbelianMonoid if R has OrderedAbelianMonoidSup

OrderedSet if R has OrderedAbelianMonoid

PartialOrder if R has OrderedAbelianMonoid

RetractableTo E

RightModule %

RightModule R

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

unitsKnown

XAlgebra R