From ab8cc85adde879fb963c94d15675783f2cf4b183 Mon Sep 17 00:00:00 2001
From: dos-reis <gdr@axiomatics.org>
Date: Tue, 14 Aug 2007 05:14:52 +0000
Subject: Initial population.

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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/algebra pgrobner.spad}
+\author{Patrizia Gianni}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{package PGROEB PolyGroebner}
+<<package PGROEB PolyGroebner>>=
+)abbrev package PGROEB PolyGroebner
+++ Author: P. Gianni
+++ Date Created: Summer 1988
+++ Date Last Updated:
+++ 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)
+       OV:=OrderedVariableList lv
+       b:L DPoly:=[pToDmp(pol)$PP for pol in lp]
+       gb:L DPoly :=groebner(b)$GroebnerPackage(F,DP,OV,DPoly)
+       [dmpToP(pp)$PP for pp in gb]
+ 
+     totalGroebner(lp: L P,lv:L E) : L P ==
+       PP:=  PolToPol(lv,F)
+       HDPoly := HomogeneousDistributedMultivariatePolynomial(lv,F)
+       HDP:=HomogeneousDirectProduct(#lv,NNI)
+       OV:=OrderedVariableList lv
+       b:L HDPoly:=[pToHdmp(pol)$PP for pol in lp]
+       gb:=groebner(b)$GroebnerPackage(F,HDP,OV,HDPoly)
+       [hdmpToP(pp)$PP for pp in gb]
+
+@
+\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 PGROEB PolyGroebner>>
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}
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