A differential analog of the Noether normalization lemma

TL;DR

This paper establishes a differential analog of the Noether normalization lemma, showing that differential algebraic varieties can be surjectively mapped onto affine space, with applications to differential equations.

Contribution

It introduces a differential version of the Noether normalization lemma, extending classical algebraic geometry results to differential algebraic varieties.

Findings

Existence of a surjective map onto affine space for differential algebraic varieties.

Construction of differentially independent generators for differential algebras.

Applications to the theory of differential equations.

Abstract

In this paper, we prove the following differential analog of the Noether normalization lemma: for every $d$-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map onto the $d$-dimensional affine space. Equivalently, for every integral differential algebra $A$ over differential field of zero characteristic there exist differentially independent $b_1, \ldots, b_d$ such that $A$ is differentially algebraic over subalgebra $B$ differentially generated by $b_1, \ldots, b_d$, and whenever $\mathfrak{p} \subset B$ is a prime differential ideal, there exists a prime differential ideal $\mathfrak{q} \subset A$ such that $\mathfrak{p} = B \cap \mathfrak{q}$. We also prove the analogous theorem for differential algebraic varieties over the ring of formal power series over an algebraically closed differential field and present some applications to differential equations.