Details:
Title  Detecting Unnecessary Reductions in an Involutive Basis Computation  Author(s)  Joachim Apel, Ralf Hemmecke  Type  Article in Journal  Abstract  We 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.~519541,
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.  Keywords  involutive basis; involutive criteria; Gröbner basis; Buchberger criteria  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/journal/07477171 
Language  English  Journal  Journal of Symbolic Computation  Volume  40  Number  45  Pages  11311149  Year  2005  Edition  0  Translation 
No  Refereed 
No  Sponsors  Austrian Science Foundation (FWF), SFB F013, project 1304 and NaturwissenschaftlichTheoretisches
Zentrum (NTZ) of the University of Leipzig 
