NormalizedTriangularSetCategory(R, E, V, P)ΒΆ

nregset.spad line 1 [edit on github]

The category of normalized triangular sets. A triangular set ts is said normalized if for every algebraic variable v of ts the polynomial select(ts, v) is normalized w.r.t. every polynomial in collectUnder(ts, v). A polynomial p is said normalized w.r.t. a non-constant polynomial q if p is constant or degree(p, mdeg(q)) = 0 and init(p) is normalized w.r.t. q. One of the important features of normalized triangular sets is that they are regular sets.

#: % -> NonNegativeInteger

from Aggregate

=: (%, %) -> Boolean

from BasicType

~=: (%, %) -> Boolean

from BasicType

algebraic?: (V, %) -> Boolean

from TriangularSetCategory(R, E, V, P)

algebraicCoefficients?: (P, %) -> Boolean

from RegularTriangularSetCategory(R, E, V, P)

algebraicVariables: % -> List V

from TriangularSetCategory(R, E, V, P)

any?: (P -> Boolean, %) -> Boolean

from HomogeneousAggregate P

augment: (List P, %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

augment: (List P, List %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

augment: (P, %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

augment: (P, List %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

autoReduced?: (%, (P, List P) -> Boolean) -> Boolean

from TriangularSetCategory(R, E, V, P)

basicSet: (List P, (P, P) -> Boolean) -> Union(Record(bas: %, top: List P), failed)

from TriangularSetCategory(R, E, V, P)

basicSet: (List P, P -> Boolean, (P, P) -> Boolean) -> Union(Record(bas: %, top: List P), failed)

from TriangularSetCategory(R, E, V, P)

coerce: % -> List P

from CoercibleTo List P

coerce: % -> OutputForm

from CoercibleTo OutputForm

coHeight: % -> NonNegativeInteger if V has Finite

from TriangularSetCategory(R, E, V, P)

collect: (%, V) -> %

from PolynomialSetCategory(R, E, V, P)

collectQuasiMonic: % -> %

from TriangularSetCategory(R, E, V, P)

collectUnder: (%, V) -> %

from PolynomialSetCategory(R, E, V, P)

collectUpper: (%, V) -> %

from PolynomialSetCategory(R, E, V, P)

construct: List P -> %

from Collection P

convert: % -> InputForm

from ConvertibleTo InputForm

copy: % -> %

from Aggregate

count: (P -> Boolean, %) -> NonNegativeInteger

from HomogeneousAggregate P

count: (P, %) -> NonNegativeInteger

from HomogeneousAggregate P

degree: % -> NonNegativeInteger

from TriangularSetCategory(R, E, V, P)

empty?: % -> Boolean

from Aggregate

empty: () -> %

from Aggregate

eq?: (%, %) -> Boolean

from Aggregate

eval: (%, Equation P) -> % if P has Evalable P

from Evalable P

eval: (%, List Equation P) -> % if P has Evalable P

from Evalable P

eval: (%, List P, List P) -> % if P has Evalable P

from InnerEvalable(P, P)

eval: (%, P, P) -> % if P has Evalable P

from InnerEvalable(P, P)

every?: (P -> Boolean, %) -> Boolean

from HomogeneousAggregate P

extend: (%, P) -> %

from TriangularSetCategory(R, E, V, P)

extend: (List P, %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

extend: (List P, List %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

extend: (P, %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

extend: (P, List %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

extendIfCan: (%, P) -> Union(%, failed)

from TriangularSetCategory(R, E, V, P)

find: (P -> Boolean, %) -> Union(P, failed)

from Collection P

first: % -> Union(P, failed)

from TriangularSetCategory(R, E, V, P)

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

headReduce: (P, %) -> P

from TriangularSetCategory(R, E, V, P)

headReduced?: % -> Boolean

from TriangularSetCategory(R, E, V, P)

headReduced?: (P, %) -> Boolean

from TriangularSetCategory(R, E, V, P)

headRemainder: (P, %) -> Record(num: P, den: R)

from PolynomialSetCategory(R, E, V, P)

iexactQuo: (R, R) -> R

from PolynomialSetCategory(R, E, V, P)

infRittWu?: (%, %) -> Boolean

from TriangularSetCategory(R, E, V, P)

initiallyReduce: (P, %) -> P

from TriangularSetCategory(R, E, V, P)

initiallyReduced?: % -> Boolean

from TriangularSetCategory(R, E, V, P)

initiallyReduced?: (P, %) -> Boolean

from TriangularSetCategory(R, E, V, P)

initials: % -> List P

from TriangularSetCategory(R, E, V, P)

internalAugment: (List P, %) -> %

from RegularTriangularSetCategory(R, E, V, P)

internalAugment: (P, %) -> %

from RegularTriangularSetCategory(R, E, V, P)

intersect: (List P, %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

intersect: (List P, List %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

intersect: (P, %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

intersect: (P, List %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

invertible?: (P, %) -> Boolean

from RegularTriangularSetCategory(R, E, V, P)

invertible?: (P, %) -> List Record(val: Boolean, tower: %)

from RegularTriangularSetCategory(R, E, V, P)

invertibleElseSplit?: (P, %) -> Union(Boolean, List %)

from RegularTriangularSetCategory(R, E, V, P)

invertibleSet: (P, %) -> List %

from RegularTriangularSetCategory(R, E, V, P)

last: % -> Union(P, failed)

from TriangularSetCategory(R, E, V, P)

lastSubResultant: (P, P, %) -> List Record(val: P, tower: %)

from RegularTriangularSetCategory(R, E, V, P)

lastSubResultantElseSplit: (P, P, %) -> Union(P, List %)

from RegularTriangularSetCategory(R, E, V, P)

latex: % -> String

from SetCategory

less?: (%, NonNegativeInteger) -> Boolean

from Aggregate

mainVariable?: (V, %) -> Boolean

from PolynomialSetCategory(R, E, V, P)

mainVariables: % -> List V

from PolynomialSetCategory(R, E, V, P)

map!: (P -> P, %) -> %

from HomogeneousAggregate P

map: (P -> P, %) -> %

from HomogeneousAggregate P

max: % -> P if P has OrderedSet

from HomogeneousAggregate P

max: ((P, P) -> Boolean, %) -> P

from HomogeneousAggregate P

member?: (P, %) -> Boolean

from HomogeneousAggregate P

members: % -> List P

from HomogeneousAggregate P

min: % -> P if P has OrderedSet

from HomogeneousAggregate P

more?: (%, NonNegativeInteger) -> Boolean

from Aggregate

mvar: % -> V

from PolynomialSetCategory(R, E, V, P)

normalized?: % -> Boolean

from TriangularSetCategory(R, E, V, P)

normalized?: (P, %) -> Boolean

from TriangularSetCategory(R, E, V, P)

parts: % -> List P

from HomogeneousAggregate P

purelyAlgebraic?: % -> Boolean

from RegularTriangularSetCategory(R, E, V, P)

purelyAlgebraic?: (P, %) -> Boolean

from RegularTriangularSetCategory(R, E, V, P)

purelyAlgebraicLeadingMonomial?: (P, %) -> Boolean

from RegularTriangularSetCategory(R, E, V, P)

purelyTranscendental?: (P, %) -> Boolean

from RegularTriangularSetCategory(R, E, V, P)

quasiComponent: % -> Record(close: List P, open: List P)

from TriangularSetCategory(R, E, V, P)

reduce: ((P, P) -> P, %) -> P

from Collection P

reduce: ((P, P) -> P, %, P) -> P

from Collection P

reduce: ((P, P) -> P, %, P, P) -> P

from Collection P

reduce: (P, %, (P, P) -> P, (P, P) -> Boolean) -> P

from TriangularSetCategory(R, E, V, P)

reduceByQuasiMonic: (P, %) -> P

from TriangularSetCategory(R, E, V, P)

reduced?: (P, %, (P, P) -> Boolean) -> Boolean

from TriangularSetCategory(R, E, V, P)

remainder: (P, %) -> Record(rnum: R, polnum: P, den: R)

from PolynomialSetCategory(R, E, V, P)

remove: (P -> Boolean, %) -> %

from Collection P

remove: (P, %) -> %

from Collection P

removeDuplicates: % -> %

from Collection P

removeZero: (P, %) -> P

from TriangularSetCategory(R, E, V, P)

rest: % -> Union(%, failed)

from TriangularSetCategory(R, E, V, P)

retract: List P -> %

from RetractableFrom List P

retractIfCan: List P -> Union(%, failed)

from RetractableFrom List P

rewriteIdealWithHeadRemainder: (List P, %) -> List P

from PolynomialSetCategory(R, E, V, P)

rewriteIdealWithRemainder: (List P, %) -> List P

from PolynomialSetCategory(R, E, V, P)

rewriteSetWithReduction: (List P, %, (P, P) -> P, (P, P) -> Boolean) -> List P

from TriangularSetCategory(R, E, V, P)

roughBase?: % -> Boolean

from PolynomialSetCategory(R, E, V, P)

roughEqualIdeals?: (%, %) -> Boolean

from PolynomialSetCategory(R, E, V, P)

roughSubIdeal?: (%, %) -> Boolean

from PolynomialSetCategory(R, E, V, P)

roughUnitIdeal?: % -> Boolean

from PolynomialSetCategory(R, E, V, P)

sample: %

from Aggregate

select: (%, V) -> Union(P, failed)

from TriangularSetCategory(R, E, V, P)

select: (P -> Boolean, %) -> %

from Collection P

size?: (%, NonNegativeInteger) -> Boolean

from Aggregate

sort: (%, V) -> Record(under: %, floor: %, upper: %)

from PolynomialSetCategory(R, E, V, P)

squareFreePart: (P, %) -> List Record(val: P, tower: %)

from RegularTriangularSetCategory(R, E, V, P)

stronglyReduce: (P, %) -> P

from TriangularSetCategory(R, E, V, P)

stronglyReduced?: % -> Boolean

from TriangularSetCategory(R, E, V, P)

stronglyReduced?: (P, %) -> Boolean

from TriangularSetCategory(R, E, V, P)

triangular?: % -> Boolean

from PolynomialSetCategory(R, E, V, P)

trivialIdeal?: % -> Boolean

from PolynomialSetCategory(R, E, V, P)

variables: % -> List V

from PolynomialSetCategory(R, E, V, P)

zeroSetSplit: (List P, Boolean) -> List %

from RegularTriangularSetCategory(R, E, V, P)

zeroSetSplit: List P -> List %

from TriangularSetCategory(R, E, V, P)

zeroSetSplitIntoTriangularSystems: List P -> List Record(close: %, open: List P)

from TriangularSetCategory(R, E, V, P)

Aggregate

BasicType

CoercibleTo List P

CoercibleTo OutputForm

Collection P

ConvertibleTo InputForm

Evalable P if P has Evalable P

finiteAggregate

HomogeneousAggregate P

InnerEvalable(P, P) if P has Evalable P

PolynomialSetCategory(R, E, V, P)

RegularTriangularSetCategory(R, E, V, P)

RetractableFrom List P

SetCategory

shallowlyMutable

TriangularSetCategory(R, E, V, P)