An Algebraic Study of Multivariable Integration and Linear Substitution

TL;DR

This paper develops an algebraic framework for multivariable integration using Rota-Baxter operators and matrix monoid actions, proposing a conjecture about the structure of operator relations.

Contribution

It introduces a novel algebraic approach to multivariable integration, combining Rota-Baxter hierarchies with linear substitutions via matrix monoids.

Findings

Constructs an operator ring acting on Rota-Baxter hierarchies

Proposes a conjecture on noncommutative Groebner basis for operator relations

Provides a theoretical foundation for algebraic multivariable integration

Abstract

We set up an algebraic theory of multivariable integration, based on a hierarchy of Rota-Baxter operators and an action of the matrix monoid as linear substitutions. Given a suitable coefficient domain with a bialgebra structure, this allows us to build an operator ring that acts naturally on the given Rota-Baxter hierarchy. We conjecture that the operator relations are a noncommutative Groebner basis for the ideal they generate.