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TitleDetecting Unnecessary Reductions in an Involutive Basis Computation
Author(s) Joachim Apel, Ralf Hemmecke
TypeArticle in Journal
AbstractWe consider the check of the involutive basis
property in a polynomial context. In order to show
that a finite generating set $F$ of a polynomial
ideal $I$ is an involutive basis one must confirm
two properties. Firstly, the set of leading terms of
the elements of $F$ has to be complete. Secondly,
one has to prove that $F$ is a Gröbner basis of $I$.
The latter is the time critical part but can be
accelerated by application of Buchberger's criteria
including the many improvements found during the
last two decades. \par Gerdt and Blinkov (Involutive
Bases of Polynomial Ideals. {\em Mathematics and
Computers in Simulation} {\bf 45}, pp.~519--541,
1998) were the first who applied these criteria in
involutive basis computations. We present criteria
which are also transferred from the theory of
Gröbner bases to involutive basis computations. We
illustrate that our results exploit the Gröbner
basis theory slightly more than those of Gerdt and
Blinkov. Our criteria apply in all cases where those
of Gerdt/Blinkov do, but we also present examples
where our criteria are superior. \par Some of our
criteria can be used also in algebras of solvable
type, \eg, Weyl algebras or enveloping algebras of
Lie algebras, in full analogy to the Gröbner basis
case. \par We show that the application of criteria
enforces the termination of the involutive basis
algorithm independent of the prolongation selection
strategy.
Keywordsinvolutive basis; involutive criteria; Gröbner basis; Buchberger criteria
ISSN0747-7171
URL http://www.sciencedirect.com/science/journal/07477171
LanguageEnglish
JournalJournal of Symbolic Computation
Volume40
Number4--5
Pages1131--1149
Year2005
Edition0
Translation No
Refereed No
SponsorsAustrian Science Foundation (FWF), SFB F013, project 1304 and Naturwissenschaftlich-Theoretisches Zentrum (NTZ) of the University of Leipzig
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