A differential analog of a theorem of Chevalley (Q2747850)

The main result of the paper is the following: Theorem 1. Let \(S\) be a differential algebra over \(\mathbb Q\) 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 zero. Then there exists a nonzero element \(a\) of \(R\) such that any homomorphism \(\varphi: R\to F\) that does not annihilate \(a\) extends to a homomorphism \(\psi:S\to F\) that does not annihilate \(b\). NEWLINENEWLINENEWLINEA consequence is: Theorem 2. Let \(F\) be a differentially closed field of characteristic zero, and let \(S\supset R\) be a differentially finitely generated differential algebra and subalgebra over \(F\). Suppose that there exists a nonzero element \(b\) of \(S\) such that any homomorphism \(\varphi:R\to F\) has only finitely many extensions \(\psi:S\to F\) satisfying \(\psi(b)\neq 0\). Then the field extension \(\operatorname {Fract}S\supset\operatorname {Fract}R\) is finite. In particular, if any homomorphism \(\varphi:R\to F\) has at most \(d\) extensions \(\psi:S\to F\) with \(\psi(b)\neq 0\), then the degree of \(\operatorname {Fract}S\) over \(\operatorname {Fract}R\) is at most \(d\).