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 (H117E1)

      Creation of Ideals and Accessing their Bases
            ideal<R | L> : RngMPolLoc, List -> RngMPolLoc
            Ideal(B) : [ RngMPolLocElt ] -> RngMPolLoc
            Ideal(f) : RngMPolLocElt -> RngMPolLoc
            IdealWithFixedBasis(B) : [ 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 ]
            Coordinates(I, f) : RngMPolLoc, RngMPolLocElt -> [ RngMPolLocElt ]
            CoordinateMatrix(I) : RngMPolLoc -> Matrix
            Example RngMPolLoc_StandardBasis (H117E2)
            Example RngMPolLoc_StandardBasis2 (H117E3)

 
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 (H117E4)

      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 (H117E5)

 
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
      Dimension(I) : RngMPolLoc -> RngIntElt, [ RngIntElt ]

 
Bibliography

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