Construction of Involutive Monomial Sets for Different Involutive Divisions

TL;DR

The paper tackles the problem of efficiently completing a finite monomial set to an involutive basis under various involutive divisions, enabling involutive-based computation of invariants like Hilbert functions. It formalizes eight divisions (including Janet and Pommaret) and introduces the MinimalInvolutiveMonomialBasis algorithm, along with a flexible Mathematica framework and a dedicated C implementation for Janet division. Key contributions include optimization strategies for separation computations, nonmultiplicative prolongations, and autoreduction, with experimental comparisons showing substantial speedups—particularly for Janet division—and demonstrations on polynomial systems. The work provides extensible tools for involutive completion in polynomial and differential systems, with practical implications for computational algebra and related applications.

Abstract

We consider computational and implementation issues for the completion of monomial sets to involution using different involutive divisions. Every of these divisions produces its own completion procedure. For the polynomial case it yields an involutive basis which is a special form of a Groebner basis, generally redundant. We also compare our Mathematica implementation of Janet division to an implementation in C.

Theorems & Definitions (10)

• definition 1
• definition 2
• remark 1
• remark 2
• remark 3
• remark 4
• remark 5
• remark 6
• remark 7
• remark 8