QEtaAlgebraBasisCategory F

qetasamba.spad line 114 [edit on github]

• F: Type

QEtaAlgebraBasisCategory(F) is a category for a data structure that can be used for reduction modulo an algebra basis (samba basis).

basis: % -> List F

basis(x) returns all basis elements.

basis: (%, Integer) -> List F

basis(x, d) returns the basis elements of grade d.

coerce: % -> OutputForm

from CoercibleTo OutputForm

initialize: List F -> %

initialize(m) creates an initial basis.

multiplier: % -> F

multiplier(x) returns multiplier element.

multiplierPower!: (%, NonNegativeInteger) -> F

multiplierPower!(x,n) returns multiplier element raised to power n. Since a power of the multiplier is needed several times, this function is mainly here to allow a domain that implements this function to cache the powers.

numberOfGaps: % -> NonNegativeInteger

numberOfGaps(x) computes reduce(+, [floor(x/n) for x in bas], 0) where t=multiplier(x), n=qetaGrade(t) and bas=basis(x). If #(basis(x,i)=1 for all i from 1 to n-1 then numberOfGaps(x) is equal to the number of gaps (see Weierstrass Gap Theorem) of the module generated by bas over C[t].

CoercibleTo OutputForm