New effective differential Nullstellensatz

TL;DR

This paper establishes new upper and lower bounds for the effective differential Nullstellensatz in characteristic zero differential fields with multiple derivations, advancing understanding beyond previous bounds that used Ackermann functions.

Contribution

It extends previous bounds to multiple derivations using model-theoretic methods, providing a more general and explicit characterization.

Findings

New bounds for differential Nullstellensatz with multiple derivations

Extension of previous bounds to broader cases

Application of model-theoretic approaches to differential algebra

Abstract

We show new upper and lower bounds for the effective differential Nullstellensatz for differential fields of characteristic zero with several commuting derivations. Seidenberg was the first to address this problem in 1956, without giving a complete solution. The first explicit bounds appeared in 2009 in a paper by Golubitsky, Kondratieva, Szanto, and Ovchinnikov, with the upper bound expressed in terms of the Ackermann function. D'Alfonso, Jeronimo, and Solern\'o, using novel ideas, obtained in 2014 a new bound if restricted to the case of one derivation and constant coefficients. To obtain the bound in the present paper without this restriction, we extend this approach and use the new methods of Freitag and Le\'on S\'anchez and of Pierce from 2014, which represent a model-theoretic approach to differential algebraic geometry.