Rolling simplexes and their commensurability. IV. (A farewell to arms!)

The text by pure algebraic reasons outlines why the spectrum of maximal ideals $\mathrm{Spec}_\mathbb{C} A$ of a countable-dimensional differential $\mathbb{C}$-algebra $A$ of transcendence degree $1$ without zero divisors is locally analytic, which means that for any $\mathbb{C}$-homomorphism $\psi_M \colon A \to \mathbb{C}$ ($M \in \mathrm{Spec}_{\mathbb C} A$) and any $a \in A$ the Taylor series $\tilde{\psi}_M (a) ={}$ $\sum\limits_{m=0}^{\infty} \psi_M(a^{(m)}) \frac{z^m}{m!}$ has nonzero radius of convergence depending on the element $a \in A$.