Effective bounds for the consistency of differential equations

TL;DR

This paper develops an effective method to determine the consistency of partial differential equations by computing bounds on prolongations, improving solution existence criteria and applications in differential algebra.

Contribution

It introduces an improved upper bound for prolongations needed to ensure realizations, advancing the algebraic approach to PDE consistency checking.

Findings

Computed an improved upper bound for prolongations

Applied bounds to characteristic sets of differential ideals

Proved a new growth result for the Hilbert-Samuel function

Abstract

One method to determine whether or not a system of partial differential equations is consistent is to attempt to construct a solution using merely the "algebraic data" associated to the system. In technical terms, this translates to the problem of determining the existence of regular realizations of differential kernels via their possible prolongations. In this paper we effectively compute an improved upper bound for the number of prolongations needed to guarantee the existence of such realizations, which ultimately produces solutions to many types of systems of partial differential equations. This bound has several applications, including an improved upper bound for the order of characteristic sets of prime differential ideals. We obtain our upper bound by proving a new result on the growth of the Hilbert-Samuel function, which may be of independent interest.