A differential Chevalley theorem

TL;DR

This paper proves a differential analog of Chevalley's theorem, extending homomorphisms in differential algebra, and explores implications for differential schemes, differentially closed fields, and related algebraic structures.

Contribution

It introduces a differential version of Chevalley's theorem, generalizes Kac's theorem, and provides new algebraic characterizations and results for differential and difference fields.

Findings

Differential Chevalley theorem proved.

Image of differential scheme under finite morphism is constructible.

Established differential Nullstellensatz and algebraic characterizations.

Abstract

We prove a differential analog of a theorem of Chevalley on extending homomorphisms for rings with commuting derivations, generalizing a theorem of Kac. As a corollary, we establish that, under suitable hypotheses, the image of a differential scheme under a finite morphism is a constructible set. We also obtain a new algebraic characterization of differentially closed fields. We show that similar results hold for differentially closed fields that are saturated, in the sense of model theory. In characteristic p > 0, we obtain related results and establish a differential Nullstellensatz. Analogs of these theorems for difference fields are also considered.