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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra gb.spad}
\author{Rudiger Gebauer, Barry Trager}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\begin{verbatim}
--------- GROEBNER PACKAGE DRAFT 06 12/01/1986
---------
--------- Example to call groebner:
---------
--------- s1:DMP[w,p,z,t,s,b]RN:= 45*p + 35*s - 165*b - 36
--------- s2:DMP[w,p,z,t,s,b]RN:= 35*p + 40*z + 25*t - 27*s
--------- s3:DMP[w,p,z,t,s,b]RN:= 15*w + 25*p*s + 30*z - 18*t - 165*b**2
--------- s4:DMP[w,p,z,t,s,b]RN:= -9*w + 15*p*t + 20*z*s
--------- s5:DMP[w,p,z,t,s,b]RN:= w*p + 2*z*t - 11*b**3
--------- s6:DMP[w,p,z,t,s,b]RN:= 99*w - 11*b*s + 3*b**2
--------- s7:DMP[w,p,z,t,s,b]RN:= b**2 + 33/50*b + 2673/10000
---------
--------- sn7:=[s1,s2,s3,s4,s5,s6,s7]
---------
--------- groebner(sn7,info)
---------
-------------------------------------------------------------------------
---------
--------- groebner -> calculate minimal Groebner Basis
---------
--------- all reductions are TOTAL reductions
---------
--------- use string " redcrit " and you get the reduced critpairs
--------- printed
---------
--------- use string " info " and you get information about
---------
--------- ci => Leading monomial for critpair calculation
--------- tci => Number of terms of polynomial i
--------- cj => Leading monomial for critpair calculation
--------- tcj => Number of terms of polynomial j
--------- c => Leading monomial of critpair polynomial
--------- tc => Number of terms of critpair polynomial
--------- rc => Leading monomial of redcritpair polynomial
--------- trc => Number of terms of redcritpair polynomial
--------- tF => Number of polynomials in reduction list F
--------- tD => Number of critpairs still to do
---------
\end{verbatim}
\section{package GB GroebnerPackage}
<<package GB GroebnerPackage>>=
)abbrev package GB GroebnerPackage
++ Authors: Gebauer, Trager
++ Date Created: 12-1-86
++ Date Last Updated: 2-28-91
++ Basic Functions: groebner normalForm
++ Related Constructors: Ideal, IdealDecompositionPackage
++ Also See:
++ AMS Classifications:
++ Keywords: groebner basis, polynomial ideal
++ References:
++ Description: \spadtype{GroebnerPackage} computes groebner
++ bases for polynomial ideals. The basic computation provides
++ a distinguished set of generators for polynomial ideals over fields.
++ This basis allows an easy test for membership: the operation \spadfun{normalForm}
++ returns zero on ideal members. When the provided coefficient domain, Dom,
++ is not a field, the result is equivalent to considering the extended
++ ideal with \spadtype{Fraction(Dom)} as coefficients, but considerably more efficient
++ since all calculations are performed in Dom. Additional argument "info" and "redcrit"
++ can be given to provide incremental information during
++ computation. Argument "info" produces a computational summary for each s-polynomial.
++ Argument "redcrit" prints out the reduced critical pairs. The term ordering
++ is determined by the polynomial type used. Suggested types include
++ \spadtype{DistributedMultivariatePolynomial},
++ \spadtype{HomogeneousDistributedMultivariatePolynomial},
++ \spadtype{GeneralDistributedMultivariatePolynomial}.
GroebnerPackage(Dom, Expon, VarSet, Dpol): T == C where
Dom: GcdDomain
Expon: OrderedAbelianMonoidSup
VarSet: OrderedSet
Dpol: PolynomialCategory(Dom, Expon, VarSet)
T== with
groebner: List(Dpol) -> List(Dpol)
++ groebner(lp) computes a groebner basis for a polynomial ideal
++ generated by the list of polynomials lp.
groebner: ( List(Dpol), String ) -> List(Dpol)
++ groebner(lp, infoflag) computes a groebner basis
++ for a polynomial ideal
++ generated by the list of polynomials lp.
++ Argument infoflag is used to get information on the computation.
++ If infoflag is "info", then summary information
++ is displayed for each s-polynomial generated.
++ If infoflag is "redcrit", the reduced critical pairs are displayed.
++ If infoflag is any other string, no information is printed during computation.
groebner: ( List(Dpol), String, String ) -> List(Dpol)
++ groebner(lp, "info", "redcrit") computes a groebner basis
++ for a polynomial ideal generated by the list of polynomials lp,
++ displaying both a summary of the critical pairs considered ("info")
++ and the result of reducing each critical pair ("redcrit").
++ If the second or third arguments have any other string value,
++ the indicated information is suppressed.
if Dom has Field then
normalForm: (Dpol, List(Dpol)) -> Dpol
++ normalForm(poly,gb) reduces the polynomial poly modulo the
++ precomputed groebner basis gb giving a canonical representative
++ of the residue class.
C== add
import OutputForm
import GroebnerInternalPackage(Dom,Expon,VarSet,Dpol)
if Dom has Field then
monicize(p: Dpol):Dpol ==
one?(lc := leadingCoefficient p) => p
inv(lc)*p
normalForm(p : Dpol, l : List(Dpol)) : Dpol ==
redPol(p,map(monicize,l))
------ MAIN ALGORITHM GROEBNER ------------------------
groebner( Pol: List(Dpol) ) ==
Pol=[] => Pol
Pol:=[x for x in Pol | x ~= 0]
Pol=[] => [0]
minGbasis(sort( degree #1 > degree #2, gbasis(Pol,0,0)))
groebner( Pol: List(Dpol), xx1: String) ==
Pol=[] => Pol
Pol:=[x for x in Pol | x ~= 0]
Pol=[] => [0]
xx1 = "redcrit" =>
minGbasis(sort( degree #1 > degree #2, gbasis(Pol,1,0)))
xx1 = "info" =>
minGbasis(sort( degree #1 > degree #2, gbasis(Pol,2,1)))
messagePrint(" ")
messagePrint("WARNING: options are - redcrit and/or info - ")
messagePrint(" you didn't type them correct")
messagePrint(" please try again")
messagePrint(" ")
[]
groebner( Pol: List(Dpol), xx1: String, xx2: String) ==
Pol=[] => Pol
Pol:=[x for x in Pol | x ~= 0]
Pol=[] => [0]
(xx1 = "redcrit" and xx2 = "info") or
(xx1 = "info" and xx2 = "redcrit") =>
minGbasis(sort( degree #1 > degree #2, gbasis(Pol,1,1)))
xx1 = "redcrit" and xx2 = "redcrit" =>
minGbasis(sort( degree #1 > degree #2, gbasis(Pol,1,0)))
xx1 = "info" and xx2 = "info" =>
minGbasis(sort( degree #1 > degree #2, gbasis(Pol,2,1)))
messagePrint(" ")
messagePrint("WARNING: options are - redcrit and/or info - ")
messagePrint(" you didn't type them correctly")
messagePrint(" please try again ")
messagePrint(" ")
[]
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<package GB GroebnerPackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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