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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra resring.spad}
\author{Patrizia Gianni}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain RESRING ResidueRing}
<<domain RESRING ResidueRing>>=
)abbrev domain RESRING ResidueRing
++ Author: P.Gianni
++ Date Created: December 1992
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: ResidueRing is the quotient of a polynomial ring by an ideal.
++ The ideal is given as a list of generators. The elements of the domain
++ are equivalence classes expressed in terms of reduced elements
ResidueRing(F,Expon,VarSet,FPol,LFPol) : Dom == Body
where
F : Field
Expon : OrderedAbelianMonoidSup
VarSet : OrderedSet
FPol : PolynomialCategory(F, Expon, VarSet)
LFPol : List FPol
Dom == Join(CommutativeRing, Algebra F) with
reduce : FPol -> $
++ reduce(f) produces the equivalence class of f in the residue ring
coerce : FPol -> $
++ coerce(f) produces the equivalence class of f in the residue ring
lift : $ -> FPol
++ lift(x) return the canonical representative of the equivalence class x
Body == add
--representation
Rep:= FPol
import GroebnerPackage(F,Expon,VarSet,FPol)
relations:= groebner(LFPol)
relations = [1] => error "the residue ring is the zero ring"
--declarations
x,y: $
--definitions
0 == 0$Rep
1 == 1$Rep
reduce(f : FPol) : $ == normalForm(f,relations)
coerce(f : FPol) : $ == normalForm(f,relations)
lift x == x :: Rep :: FPol
x + y == x +$Rep y
-x == -$Rep x
x*y == normalForm(lift(x *$Rep y),relations)
(n : Integer) * x == n *$Rep x
(a : F) * x == a *$Rep x
x = y == x =$Rep y
characteristic == characteristic$F
coerce(x) : OutputForm == coerce(x)$Rep
@
\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>>
<<domain RESRING ResidueRing>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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