From analytical mechanical problems to rewriting theory through M. Janet

TL;DR

This paper explores the historical development and deep connections between Gr"obner basis theory for D-modules and linear rewriting theory, highlighting Janet's contributions to algebraic analysis of PDE systems.

Contribution

It presents generalizations of Janet's involutive division and their relations to Gr"obner basis theory, enriching the algebraic understanding of linear PDE systems.

Findings

Historical link between Gr"obner bases and rewriting theory

Generalizations of Janet's involutive division

Connections to algebraic analysis of PDEs

Abstract

This note surveys the historical background of the Gr\"obner basis theory for D-modules and linear rewriting theory. The objective is to present a deep interaction of these two fields largely developed in algebra throughout the twentieth century. We recall the work of M. Janet on the algebraic analysis on linear partial differential systems that leads to the notion of involutive division. We present some generalizations of the division introduced by M. Janet and their relations with Gr\"obner basis theory.