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TitleInvolutive Bases of Polynomial Ideals
Author(s) Yuri A. Blinkov, Vladimir P. Gerdt
TypeArticle in Journal
Abstract"In this paper we consider an algorithmic technique
more general than that proposed by Zharkov and Blinkov
for the involutive analysis of polynomial ideals. It is
based on a new concept of involutive monomial division
which is defined for a monomial set. Such a division
provides for each monomial the self-consistent
separation of the whole set of variables into two
disjoint subsets. They are called multiplicative and
non-multiplicative. Given an admissible ordering, this
separation is applied to polynomials in terms of their
leading monomials. As special cases of the separation
we consider those introduced by Janet, Thomas and
Pommaret for the purpose of algebraic analysis of
partial differential equations. Given involutive
division, we define an involutive reduction and an
involutive normal form. Then we introduce, in terms of
the latter, the concept of involutivity for polynomial
systems. We prove that an involutive system is a
special, generally redundant, form of a Groebner basis.
An algorithm for construction of involutive bases is
proposed. It is shown that involutive divisions
satisfying certain conditions, for example, those of
Janet and Thomas, provide an algorithmic construction
of an involutive basis for any polynomial ideal. Some
optimization in computation of involutive bases is also
analyzed. In particular, we incorporate Buchberger's
chain criterion to avoid unnecessary reductions. The
implementation for Pommaret division has been done in
JournalMathematics and Computers in Simulation
Translation No
Refereed No