Combinatorics of involutive divisions

TL;DR

This paper explores the combinatorial properties of involutive divisions on sets of terms, introducing new concepts like relative involutive divisions and graph tools to analyze ideal membership and structure.

Contribution

It extends Janet's classical involutive division theory by defining relative involutive divisions and developing graph-based methods for analyzing ideal membership.

Findings

Introduces the concept of relative involutive divisions.

Develops two graph tools related to involutive divisions.

Provides methods to identify generators and elements in ideals and escalier.

Abstract

The classical involutive division theory by Janet decomposes in the same way both the ideal and the escalier. The aim of this paper, following Janet's approach, is to discuss the combinatorial properties of involutive divisions, when defined on the set of all terms in a fixed degree D, postponing the discussion of ideal membership and related test. We adapt the theory by Gerdt and Blinkov, introducing relative involutive divisions and then, given a complete description of the combinatorial structure of a relative involutive division, we turn our attention to the problem of membership. In order to deal with this problem, we introduce two graphs as tools, one is strictly related to Seiler's L-graph, whereas the second generalizes it, to cover the case of "non-continuous" (in the sense of Gerdt-Blinkov) relative involutive divisions. Indeed, given an element in the ideal (resp. escalier), walking backwards (resp. forward) in the graph, we can identify all the other generators of the ideal (resp. elements of degree D in the escalier).