pgrobner.spad

)abbrev package PGROEB PolyGroebner
++ Author: P. Gianni
++ Date Created: Summer 1988
++ Basic Functions:
++ Related Constructors: GroebnerPackage
++ Also See:
++ AMS Classifications:
++ Keywords: groebner basis, polynomial ideals
++ References:
++ Description:
++ Groebner functions for P F
++   This package is an interface package to the groebner basis
++ package which allows you to compute groebner bases for polynomials
++ in either lexicographic ordering or total degree ordering refined
++ by reverse lex. The input is the ordinary polynomial type which
++ is internally converted to a type with the required ordering.
++ The resulting grobner basis is converted back to ordinary polynomials.
++ The ordering among the variables is controlled by an explicit list
++ of variables which is passed as a second argument. The coefficient
++ domain is allowed to be any gcd domain, but the groebner basis is
++ computed as if the polynomials were over a field.

PolyGroebner(F) : C == T

 where
  F      :   GcdDomain
  NNI    ==>  NonNegativeInteger
  P      ==>  Polynomial F
  L      ==>  List
  E      ==>  Symbol

  C == with
     lexGroebner   :    (L P, L E)  ->  L P
       ++ lexGroebner(lp, lv) computes Groebner basis
       ++ for the list of polynomials lp in lexicographic order.
       ++ The variables are ordered by their position in the list lv.

     totalGroebner :    (L P, L E) ->  L P
       ++ totalGroebner(lp, lv) computes Groebner basis
       ++ for the list of polynomials lp with the terms
       ++ ordered first by total degree and then
       ++ refined by reverse lexicographic ordering.
       ++ The variables are ordered by their position in the list lv.

  T == add
     lexGroebner(lp : L P, lv : L E) : L P ==
       PP :=  PolToPol(lv, F)
       DPoly := DistributedMultivariatePolynomial(lv, F)
       DP := DirectProduct(#lv, NNI)
       b : L DPoly := [pToDmp(pol)$PP for pol in lp]
       gb : L DPoly := groebner(b)$GroebnerPackage(F, DP, DPoly)
       [dmpToP(pp)$PP for pp in gb]

     totalGroebner(lp : L P, lv : L E) : L P ==
       PP :=  PolToPol(lv, F)
       HDPoly := HomogeneousDistributedMultivariatePolynomial(lv, F)
       HDP := HomogeneousDirectProduct(#lv, NNI)
       b : L HDPoly := [pToHdmp(pol)$PP for pol in lp]
       gb := groebner(b)$GroebnerPackage(F, HDP, HDPoly)
       [hdmpToP(pp)$PP for pp in gb]

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