A Bound for Orders in Differential Nullstellensatz

TL;DR

This paper establishes the first known bounds on the order of differentiations needed in the differential Nullstellensatz for both partial and ordinary differential equations, filling a longstanding gap in the field.

Contribution

It provides the first complete bounds for differentiation orders in differential Nullstellensatz, extending previous partial results and complementing algebraic geometry bounds.

Findings

First known bounds for differentiation orders in differential Nullstellensatz

Applicable to both partial and ordinary differential equations

Fills a gap left by previous incomplete solutions

Abstract

We give the first known bound for orders of differentiations in differential Nullstellensatz for both partial and ordinary algebraic differential equations. This problem was previously addressed by A. Seidenberg but no complete solution was given. Our result is a complement to the corresponding result in algebraic geometry, which gives a bound on degrees of polynomial coefficients in effective Nullstellensatz.