Non-affine differential-algebraic curves do not exist
The paper outlines why the spectrum of maximal ideals ${\rm Spec}_\mathbb{C} A$ of a countably-dimensional differential $\mathbb{C}$-algebra $A$ of transcendence degree 1 without zero devisors is locally analytic, which means that for any $\mathbb{C}$-homomorphism $\psi_M : A \to \mathbb{C}$ ($M \in {\rm Spec}_{\mathbb{C}} A$) and any $a \in A$ the Taylor series $\widetilde{\psi}_M (a) \stackrel{{\rm def}}{=} \sum\limits_{m=0}^{\infty} \psi_M(a^{(m)}) \frac{z^m}{m!}$ has nonzero radius of convergence depending on the element $a \in A$.