Toward involutive bases over effective rings
The origins of the Janet-Riquier theory of involutive divisions come from a paper of \textit{M. Janet} [C. R. Acad. Sci., Paris 170, 1101--1103 (1920; JFM 47.0440.03)] and gave rise to the notion of involutive basis. In that paper the author developed an algebraic theory that turned out to be useful for the study of linear systems of partial differential equations, also from a computational symbolic point of view. Then, the idea of involutive division have been further improved by several authors also in other frameworks. For example, recently, involutive divisions have been exploited in order to develop computational methods for the study of Hilbert schemes in a free-term order setting (see [\textit{M. Ceria} et al., J. Symb. Comput. 68, Part 2, 87--108 (2015; Zbl 1311.13036); \textit{M. Ceria} et al., J. Symb. Comput. 95, 100--133 (2019; Zbl 1430.13043)] and the reference therein). Today it is established that Janet's studies anticipated the ideas of Gröbner bases theory, in particular with the notions of weak and strong involutive bases (see [\textit{W. M. Seiler}, Appl. Algebra Eng. Commun. Comput. 20, No. 3--4, 207--259 (2009; Zbl 1175.13012); Appl. Algebra Eng. Commun. Comput. 20, No. 3--4, 261--338 (2009; Zbl 1175.13011)]). Indeed, the notion of involutive divisor gives rice to a decomposition of the set of terms of a monomial ideal, which can be detected by involutive bases and is a partition in the strong case. In this context, the authors of the paper under review focus their attention on the very interesting problem that consists in making all the theory to be applicable to monomials with coefficients over an effective ring. As it is clearly pointed out in the Introduction, with this aim in mind, four important goals can be considered: (1) to adapt the detection of involutive divisors over a field to monomials over an effective ring, if possible; (2) to investigate the case of strong involutive bases; (3) to reformulate the algorithm of W. Seiler for computing weak involutive bases; (4) to extend involutivity to the Noetherian setting. The main results that are achieved in this paper are: \begin{itemize} \item a complete affirmative answer to problem~(1) \item an outline of solution of problem (3) \item a conjecture about the computation of strong involutive bases. \end{itemize} Some explicative examples also are exhibited.
scientific article; zbMATH DE number 7348008
scientific article