The rationality–transcendental transcendence dichotomy for affine varieties

Let $K$ be an algebraically closed field of characteristic zero, let $V\subseteq\mathbb{A}^d_K$ be an affine variety, and define its generating power series by

$$F_V(x_1,\ldots,x_d):=\sum_{(n_1,\ldots,n_d)\in V\cap\mathbb{N}^d}x_1^{n_1}\cdots x_d^{n_d}.$$

Rationality–transcendence conjecture. The power series $F_V(x_1,\ldots,x_d)$ is either rational or transcendentally transcendental.

This is proposed as an analogue of the Pólya–Carlson theorem in the context of algebraic geometry and differential algebra. The claim excludes an intermediate possibility in which the generating series is transcendental but not transcendentally transcendental.