Effective uniform bounding in partial differential fields

TL;DR

This paper establishes effective bounds for the size of solution sets and their Zariski closures in partial differential fields, extending previous results from ordinary differential equations to the multivariate case.

Contribution

It provides explicit upper bounds for solution set sizes and degrees of Zariski closures in partial differential fields with commuting derivations, advancing the understanding of solution complexity.

Findings

Upper bounds for finite solution set sizes

Bounds for degrees of Zariski closures

Methods applicable to systems with multiple derivations

Abstract

Motivated by the effective bounds of ordinary differential equations, we prove an effective version of uniform bounding for partial differential fields with commuting derivations. More precisely, we provide an upper bound for the size of finite solution sets of partial differential polynomial equations in terms of data explicitly given in the equations and independent of parameters. Our methods also produce an upper bound for the degree of the Zariski closure of solution sets, whether they are finite or not.