A Differential Analog of the Noether Normalization Lemma

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 on to 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.