Nonaffine differential-algebraic curves do not exist

Abstract

The paper outlines why the spectrum of maximal ideals Spec_ℂA of a countable-dimensional differential ℂ-algebra A of transcendence degree 1 without zero divisors is locally analytic, which means that for any ℂ-homomorphism ψ_M: A → ℂ (M ∈ Spec_ℂA) and any a ∈ A the Taylor series \(\widetilde {{\psi _M}}{\left( a \right)^{\underline{\underline {def}} }}\sum\limits_{m = 0}^\infty {\psi M\left( {{a^{\left( m \right)}}} \right)} \frac{{{z^m}}}{{m!}}\) has nonzero radius of convergence depending on the element a ∈ A.