Minimal involutive bases

Abstract

In this paper we present an algorithm for construction of minimal involutive polynomial bases which are Gröbner bases of the special form. The most general involutive algorithms are based on the concept of involutive monomial division which leads to partition of variables into multiplicative and non-multiplicative. This partition gives thereby the self-consistent computational procedure for constructing an involutive basis by performing non-multiplicative prolongations and multiplicative reductions. Every specific involutive division generates a particular form of involutive computational procedure. In addition to three involutive divisions used by Thomas, Janet and Pommaret for analysis of partial differential equations we define two new ones. These two divisions, as well as Thomas division, do not depend on the order of variables. We prove noetherity, continuity and constructivity of the new divisions that provides correctness and termination of involutive algorithms for any finite set of input polynomials and any admissible monomial ordering. We show that, given an admissible monomial ordering, a monic minimal involutive basis is uniquely defined and thereby can be considered as canonical much like the reduced Gröbner basis.

Involutive Bases in the Weyl Algebra

“…Within the computer algebra system MuPAD † we implemented the involutive completion algorithm (based on the optimized form proposed by Gerdt and Blinkov, 1998b) for ideals in arbitrary polynomial algebras of solvable type (in the generalized sense of Seiler, 2001). For the special case of the Weyl algebra, this implementation provides tools for the (de)homogenization of polynomials and for the lift of both multiplicative monomial orders and involutive divisions.…”

“…Like for Buchberger's algorithm, a number of optimizations have been found (their effect is less profound, as one can show that many of the optimizations found for Buchberger's algorithm are automatically included in the involutive strategy). In the examples in Section 6 we will always use a more sophisticated form of the algorithm given by Gerdt and Blinkov (1998b), as it typically leads to smaller bases. This concerns in particular the Janet division which we will mainly use.…”

Construction of Involutive Monomial Sets for Different Involutive Divisions

“…Some of them are applicable to different divisions, others are concerned with the completion procedure in general. The basic operations on monomial sets are the same for the computation of involutive bases of polynomial [3,4] and differential systems [9], so the improvements described here are relevant for these computations.…”

“…Besides, we shortly describe an implementation of Janet division in C and compare the running times for both implementations. Though in this paper we consider involutivity of monomial ideals, all the underlying operations with involutive divisions and monomials enter in more general completion procedures for polynomial [3,4] and differential systems [9].…”

Janet?s approach to presentations and resolutions for polynomials and linear pdes

“…V. Gerdt and collaborators have shown that Janet's constructive ideas lead to very effective method cf. [8], [9], [23], [24]. They created an axiomatic framework for Janet's approach called involutive division and developed very efficient involutive division algorithms.…”