Nonzero orders of a function occur at only finitely many normalized places

Let $E/\mathbb{C}$ be a finitely generated field extension of transcendence degree one, and let $f\in E$. Among the normalized discrete places of $E$ that are trivial on $\mathbb{C}^\times$ (the type PolynomialCurvePlace), only finitely many places $\nu$ satisfy $\operatorname{ord}_\nu(f)\neq 0$.

This is the standard finiteness of support for a principal divisor on a smooth projective model of a one-variable function field over an algebraically closed constant field. The zero function is assigned order zero at every place by the untop₀ convention, so its support is empty.