RegularChain(R, ls)

zerodim.spad line 79 [edit on github]

• R : GcdDomain
• ls : List(Symbol)

A domain for regular chains (i.e. regular triangular sets) over a Gcd-Domain and with a fix list of variables. This is just a front-end for the RegularTriangularSet domain constructor.

# : % -> NonNegativeInteger

from Aggregate

= : (%, %) -> Boolean

from BasicType

algebraic? : (OrderedVariableList(ls), %) -> Boolean

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

algebraicCoefficients? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Boolean

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

algebraicVariables : % -> List(OrderedVariableList(ls))

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

any? : (Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> Boolean

from HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

augment : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

augment : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), List(%)) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

augment : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

augment : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), List(%)) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

autoReduced? : (%, Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))))) -> Boolean

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

basicSet : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))) -> Union(Record(bas : %, top : List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))), "failed")

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

basicSet : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))) -> Union(Record(bas : %, top : List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))), "failed")

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

coHeight : % -> NonNegativeInteger

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

coerce : % -> List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

from CoercibleTo(List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))))

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

collect : (%, OrderedVariableList(ls)) -> %

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

collectQuasiMonic : % -> %

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

collectUnder : (%, OrderedVariableList(ls)) -> %

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

collectUpper : (%, OrderedVariableList(ls)) -> %

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

construct : List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))) -> %

from Collection(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

convert : % -> InputForm if NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)) has ConvertibleTo(InputForm)

from ConvertibleTo(InputForm)

copy : % -> %

from Aggregate

count : (Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> NonNegativeInteger

from HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

count : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> NonNegativeInteger

from HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

degree : % -> NonNegativeInteger

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

empty : () -> %

from Aggregate

empty? : % -> Boolean

from Aggregate

eq? : (%, %) -> Boolean

from Aggregate

eval : (%, Equation(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))) -> % if NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)) has Evalable(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

from Evalable(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

eval : (%, List(Equation(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))))) -> % if NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)) has Evalable(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

from Evalable(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

eval : (%, List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))) -> % if NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)) has Evalable(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

from InnerEvalable(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

eval : (%, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))) -> % if NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)) has Evalable(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

from InnerEvalable(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

every? : (Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> Boolean

from HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

extend : (%, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))) -> %

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

extend : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

extend : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), List(%)) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

extend : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

extend : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), List(%)) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

extendIfCan : (%, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))) -> Union(%, "failed")

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

find : (Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> Union(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), "failed")

from Collection(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

first : % -> Union(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), "failed")

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

headReduce : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

headReduced? : % -> Boolean

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

headReduced? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Boolean

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

headRemainder : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Record(num : NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), den : R)

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

iexactQuo : (R, R) -> R

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

infRittWu? : (%, %) -> Boolean

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

initiallyReduce : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

initiallyReduced? : % -> Boolean

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

initiallyReduced? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Boolean

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

initials : % -> List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

internalAugment : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> %

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

internalAugment : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> %

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

intersect : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

intersect : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), List(%)) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

intersect : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

intersect : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), List(%)) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

invertible? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Boolean

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

invertible? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> List(Record(val : Boolean, tower : %))

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

invertibleElseSplit? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Union(Boolean, List(%))

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

invertibleSet : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

last : % -> Union(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), "failed")

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

lastSubResultant : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> List(Record(val : NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), tower : %))

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

lastSubResultantElseSplit : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Union(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), List(%))

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

latex : % -> String

from SetCategory

less? : (%, NonNegativeInteger) -> Boolean

from Aggregate

mainVariable? : (OrderedVariableList(ls), %) -> Boolean

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

mainVariables : % -> List(OrderedVariableList(ls))

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

map : (Mapping(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> %

from HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

map! : (Mapping(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> %

from HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

max : % -> NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)) if NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)) has OrderedSet

from HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

max : (Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))

from HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

member? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Boolean

from HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

members : % -> List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

from HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

min : % -> NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)) if NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)) has OrderedSet

from HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

more? : (%, NonNegativeInteger) -> Boolean

from Aggregate

mvar : % -> OrderedVariableList(ls)

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

normalized? : % -> Boolean

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

normalized? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Boolean

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

parts : % -> List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

from HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

purelyAlgebraic? : % -> Boolean

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

purelyAlgebraic? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Boolean

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

purelyAlgebraicLeadingMonomial? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Boolean

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

purelyTranscendental? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Boolean

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

quasiComponent : % -> Record(close : List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), open : List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))))

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

reduce : (Mapping(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))

from Collection(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

reduce : (Mapping(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))) -> NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))

from Collection(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

reduce : (Mapping(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))) -> NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))

from Collection(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

reduce : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %, Mapping(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))) -> NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

reduceByQuasiMonic : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

reduced? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %, Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))) -> Boolean

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

remainder : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Record(rnum : R, polnum : NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), den : R)

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

remove : (Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> %

from Collection(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

remove : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> %

from Collection(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

removeDuplicates : % -> %

from Collection(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

removeZero : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

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

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

retract : List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))) -> %

from RetractableFrom(List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))))

retractIfCan : List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))) -> Union(%, "failed")

from RetractableFrom(List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))))

rewriteIdealWithHeadRemainder : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

rewriteIdealWithRemainder : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

rewriteSetWithReduction : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %, Mapping(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))) -> List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

roughBase? : % -> Boolean

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

roughEqualIdeals? : (%, %) -> Boolean

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

roughSubIdeal? : (%, %) -> Boolean

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

roughUnitIdeal? : % -> Boolean

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

sample : () -> %

from Aggregate

select : (Mapping(Boolean, NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), %) -> %

from Collection(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

select : (%, OrderedVariableList(ls)) -> Union(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), "failed")

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

size? : (%, NonNegativeInteger) -> Boolean

from Aggregate

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

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

squareFreePart : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> List(Record(val : NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), tower : %))

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

stronglyReduce : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

stronglyReduced? : % -> Boolean

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

stronglyReduced? : (NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), %) -> Boolean

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

triangular? : % -> Boolean

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

trivialIdeal? : % -> Boolean

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

variables : % -> List(OrderedVariableList(ls))

from PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

zeroSetSplit : List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))) -> List(%)

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

zeroSetSplit : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), Boolean) -> List(%)

from RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

zeroSetSplit : (List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))), Boolean, Boolean) -> List(%)

zeroSetSplit(lp, clos?, info?) returns a list lts of regular chains such that the union of the closures of their regular zero sets equals the affine variety associated with lp. Moreover, if clos? is false then the union of the regular zero set of the ts (for ts in lts) equals this variety. If info? is true then some information is displayed during the computations. See zeroSetSplit.

zeroSetSplitIntoTriangularSystems : List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))) -> List(Record(close : %, open : List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))))

from TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

~= : (%, %) -> Boolean

from BasicType

InnerEvalable(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

PolynomialSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

ConvertibleTo(InputForm)

Aggregate

SetCategory

Collection(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

BasicType

CoercibleTo(List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))))

RetractableFrom(List(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls))))

TriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

CoercibleTo(OutputForm)

Evalable(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

HomogeneousAggregate(NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))

finiteAggregate

shallowlyMutable

RegularTriangularSetCategory(R, IndexedExponents(OrderedVariableList(ls)), OrderedVariableList(ls), NewSparseMultivariatePolynomial(R, OrderedVariableList(ls)))