@**article**{RISC2492,author = {Joachim Apel and Ralf Hemmecke},

title = {{Detecting Unnecessary Reductions in an Involutive Basis Computation}},

language = {english},

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 Gr�bner
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.
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
Gr�bner bases to involutive basis computations. We illustrate that our
results exploit the Gr�bner 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.
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 Gr�bner basis case.
We show that the application of criteria enforces the termination of
the involutive basis algorithm independent of the prolongation
selection strategy.
},

journal = {Journal of Symbolic Computation},

volume = {40},

number = {4--5},

pages = {1131--1149},

isbn_issn = {ISSN 0747-7171},

year = {2005},

refereed = {yes},

sponsor = {FWF SFB F013, project 1304; Naturwissenschaftlich-Theoretisches Zentrum (NTZ) of the University of Leipzig, Germany},

length = {19},

url = {http://www.sciencedirect.com/science/journal/07477171}

}