Macaulay inverse systems revisited

TL;DR

This paper revisits Macaulay's 1916 work on inverse systems, providing a comprehensive explanation within modern algebraic analysis and PDE theory, with detailed examples and computational insights.

Contribution

It offers the first complete modern reinterpretation of Macaulay's inverse systems chapter using algebraic analysis and PDE frameworks, bridging historical and contemporary methods.

Findings

Full explanation of inverse systems within algebraic analysis.

Explicit examples illustrating the theory.

Guidelines for developing computer algebra packages.

Abstract

Since its original publication in 1916 under the title "The Algebraic Theory of Modular Systems", the book by F. S. Macaulay has attracted a lot of scientists with a view towards pure mathematics (D. Eisenbud,...) or applications to control theory (U. Oberst,...).However, a carefull examination of the quotations clearly shows that people have hardly been looking at the last chapter dealing with the so-called "inverse systems", unless in very particular situations. The purpose of this paper is to provide for the first time the full explanation of this chapter within the framework of the formal theory of systems of partial differential equations (Spencer operator on sections, involution,...) and its algebraic counterpart now called "algebraic analysis" (commutative and homological algebra, differential modules,...). Many explicit examples are fully treated and hints are given towards the way to work out computer algebra packages.