Minimal Involutive Bases
Vladimir P. Gerdt, Yuri A. Blinkov
TL;DR
The paper addresses the computation of minimal involutive bases, a canonical form of Groebner bases, by leveraging involutive divisions that partition variables into multiplicative and non-multiplicative sets via a division $L$. It extends the framework with five divisions (Thomas, Janet, Pommaret, Division I, Division II), proves fundamental properties (noetherity, continuity, constructivity), and shows that a monic minimal $L$-involutive basis is unique for a fixed admissible ordering. It introduces two constructive algorithms, $MinimalInvolutiveMonomialBasis$ and $MinimalInvolutiveBasis$, to obtain minimal bases efficiently and to reduce the number of intermediate polynomials, while still yielding a Gröbner basis when termination is not guaranteed. The work extends to polynomial ideals and hints at applications to PDE-like differential systems, offering canonical, efficient alternatives to Buchberger-type methods and potential generalizations to other algebraic settings.
Abstract
In this paper we present an algorithm for construction of minimal involutive polynomial bases which are Groebner 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 Groebner basis.