Algorithmic Thomas Decomposition of Algebraic and Differential Systems

TL;DR

This paper introduces a new algorithm for decomposing algebraic and differential systems into simple, disjoint subsystems, improving solution partitioning and analysis of complex equations.

Contribution

It develops a novel algorithm based on Thomas's ideas for disjoint decomposition of algebraic and differential systems, with proofs and implementation details.

Findings

Algorithm successfully decomposes systems into simple subsystems

Implementation in Maple demonstrates practical efficiency

Comparison shows advantages over existing methods

Abstract

In this paper, we consider systems of algebraic and non-linear partial differential equations and inequations. We decompose these systems into so-called simple subsystems and thereby partition the set of solutions. For algebraic systems, simplicity means triangularity, square-freeness and non-vanishing initials. Differential simplicity extends algebraic simplicity with involutivity. We build upon the constructive ideas of J. M. Thomas and develop them into a new algorithm for disjoint decomposition. The given paper is a revised version of a previous paper and includes the proofs of correctness and termination of our decomposition algorithm. In addition, we illustrate the algorithm with further instructive examples and describe its Maple implementation together with an experimental comparison to some other triangular decomposition algorithms.