Nonstandard methods for bounds in differential polynomial rings

TL;DR

This paper extends nonstandard methods to differential polynomial rings to address bounds on degrees and orders, providing partial solutions to primality and related problems in differential algebra.

Contribution

It introduces nonstandard techniques to differential polynomial rings, offering new bounds and equivalences for primality and the differential Nullstellensatz.

Findings

Partial answer to the primality problem

Equivalence of primality with Ritt problem

Existence of bounds for characteristic sets and Nullstellensatz

Abstract

Motivated by the problem of the existence of bounds on degrees and orders in checking primality of radical (partial) differential ideals, the nonstandard methods of van den Dries and Schmidt ["Bounds in the theory of polynomial rings over fields. A nonstandard approach.", Inventionnes Mathematicae, 76:77--91, 1984] are here extended to differential polynomial rings over differential fields. Among the standard consequences of this work are: a partial answer to the primality problem, the equivalence of this problem with several others related to the Ritt problem, and the existence of bounds for characteristic sets of minimal prime differential ideals and for the differential Nullstellensatz.