New order bounds in differential elimination algorithms

TL;DR

This paper introduces a new upper bound for the derivatives' orders in the Rosenfeld-Groebner algorithm, enhancing the analysis of differential polynomial systems with multiple derivations.

Contribution

It provides the first general order bound for systems with multiple derivations, extending previous single-derivation results using antichain sequence bounds.

Findings

New order bound for derivatives in differential elimination

Applicable to systems with arbitrary commuting derivations

Improves understanding of differential ideal decomposition

Abstract

We present a new upper bound for the orders of derivatives in the Rosenfeld-Groebner algorithm. This algorithm computes a regular decomposition of a radical differential ideal in the ring of differential polynomials over a differential field of characteristic zero with an arbitrary number of commuting derivations. This decomposition can then be used to test for membership in the given radical differential ideal. In particular, this algorithm allows us to determine whether a system of polynomial PDEs is consistent. Previously, the only known order upper bound was given by Golubitsky, Kondratieva, Moreno Maza, and Ovchinnikov for the case of a single derivation. We achieve our bound by associating to the algorithm antichain sequences whose lengths can be bounded using the results of Leon Sanchez and Ovchinnikov.