Introduction
Elements and Local Monomial Orders
Local Lexicographical: llex
Local Graded Lexicographical: lglex
Local Graded Reverse Lexicographical: lgrev-lex
Local Polynomial Rings and Ideals
Creation of Local Polynomial Rings and Accessing their Monomial Orders LocalPolynomialRing(K, n) : Rng, RngIntElt -> RngMPolLoc LocalPolynomialRing(K, n, order) : Rng, RngIntElt, MonStgElt, ... -> RngMPolLoc LocalPolynomialRing(K, n, T) : Rng, RngIntElt, Tup -> RngMPolLoc MonomialOrder(R) : RngMPolLoc -> Tup MonomialOrderWeightVectors(R) : RngMPol -> [ [ FldRatElt ] ] Localization(R) : RngMPol -> RngMPolLoc Example RngMPolLoc_Order (H99E1)
Creation of Ideals and Accessing their Bases ideal<R | L> : RngMPolLoc, List -> RngMPolLoc Ideal(B) : [ RngMPolLocElt ] -> RngMPolLoc Ideal(f) : RngMPolLocElt -> RngMPolLoc Basis(I) : RngMPolLoc -> [ RngMPolLocElt ] BasisElement(I, i) : RngMPolLoc, RngIntElt -> RngMPolLocElt
Standard Bases
Construction of Standard Bases StandardBasis(I) : RngMPolLoc -> RngMPolLocElt StandardBasis(S) : [ RngMPolLocElt ] -> [ RngMPolLocElt ] Example RngMPolLoc_StandardBasis (H99E2) Example RngMPolLoc_StandardBasis2 (H99E3)
Operations on Ideals
Basic Operations I + J : RngMPolLoc, RngMPolLoc -> RngMPolLoc I * J : RngMPolLoc, RngMPolLoc -> RngMPolLoc I ^ k : RngMPolLoc, RngIntElt -> RngMPolLoc QuotientDimension(I) : RngMPol -> RngIntElt Generic(I) : RngMPolLoc -> RngMPolLoc LeadingMonomialIdeal(I) : RngMPolLoc -> RngMPolLoc I meet J : RngMPolLoc, RngMPolLoc -> RngMPolLoc &meet S : [ RngMPolLoc ] -> RngMPolLoc
Ideal Predicates I eq J : RngMPolLoc, RngMPolLoc -> BoolElt I ne J : RngMPolLoc, RngMPolLoc -> BoolElt I notsubset J : RngMPolLoc, RngMPolLoc -> BoolElt I subset J : RngMPolLoc, RngMPolLoc -> BoolElt IsZero(I) : RngMPolLoc -> BoolElt IsProper(I) : RngMPolLoc -> BoolElt IsZeroDimensional(I) : RngMPolLoc -> BoolElt Example RngMPolLoc_IdealArithmetic (H99E4)
Operations on Elements of Ideals f in I : RngMPolLocElt, RngMPolLoc -> BoolElt NormalForm(f, I) : RngMPolLocElt, RngMPolLoc -> RngMPolLocElt f notin I : RngMPolLocElt, RngMPolLoc -> BoolElt Example RngMPolLoc_ElementOperations (H99E5)
Changing Coefficient Ring ChangeRing(I, L) : RngMPolLoc, Rng -> RngMPolLoc
Changing Monomial Order ChangeOrder(I, Q) : RngMPolLoc, RngMPolLoc -> RngMPolLoc, Map ChangeOrder(I, order) : RngMPolLoc, ..., -> RngMPolLoc, Map
Dimension of Ideals
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