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LOCAL POLYNOMIAL RINGS

 
Acknowledgements
 
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
      Creation of Ideals and Accessing their Bases
 
Standard Bases
      Construction of Standard Bases
 
Operations on Ideals
      Basic Operations
      Ideal Predicates
      Operations on Elements of Ideals
 
Changing Coefficient Ring
 
Changing Monomial Order
 
Dimension of Ideals
 
Bibliography







DETAILS

 
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 (H107E1)

      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 (H107E2)
            Example RngMPolLoc_StandardBasis2 (H107E3)

 
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 (H107E4)

      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 (H107E5)

 
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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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012