A differential analog of a theorem of Chevalley

In this note a proof of a differential analog of Chevalley’s theorem [C] on homomorphism extensions is given. An immediate corollary is a condition of finitenes of extensions of differential algebras and several equivalent definitions of a differentially closed field, including Kolchin’s Nullstellensatz. In this note I give a proof of the following differential analog of Chevalley’s theorem [C] on homomorphism extensions. Theorem 1. Let S be a differential algebra over Q with no zero divisors and let b be a nonzero element of S. Let R be a differential subalgebra of S over which S is differentially finitely generated. Let F be a differentially closed field of characteristic 0. Then there exists a non-zero element a of R such that any homomorphism φ : R → F which does not annihilate a extends to a homomorphism ψ : S → F which does not annihilate b. An almost immediate consequence of the proof of Theorem 1 is Theorem 2. Let F be a differentially closed field of characteristic 0 and let S ⊃ R be differentially finitely generated differential algebra and subalgebra over F . Suppose that there exists a non-zero element b of S such that any homomorphism φ : R → F has only finitely many extensions ψ : S → F satisfying ψ(b) 6= 0. Then the field extension Fract S ⊃ Fract R is finite. In particular, if any homomorphism φ : R → F has at most d extensions ψ : S → F with ψ(b) 6= 0, then the degree of Fract S over Fract R is at most d. An immediate corollary of Theorem 1 is Kolchin’s Nullstellensatz [K] and its earlier weaker versions by Ritt [R], Cohn [Cohn] and Seidenberg [S]. As far as I can understand it, Theorem 1 is closely related to Blum’s elimination of quantifiers theorem [Blum], [M] in the model theory of differentially closed fields. In Section 1 I explain the necessary background on Differential Algebra, in Sections 2 and 3 give proofs of Theorems 1 and 2 and in Section 4 give about a dozen of equivalent definitions of Department of Mathematics, M.I.T., Cambridge, MA 02139, USA. [email protected] Supported in part by NSF grant DMS-9970007.