Automatic Meromorphy in Non-Archimedean Geometry

Abstract In this article, we prove that if $U$ is a Zariski-open subset of a reduced non-Archimedean analytic space $X$ and $f$ is an analytic function on $U$ whose zero-locus is equal to $Z\cap U$ for some Zariski-closed susbet $Z$ of $X$, then $f$ extends to a meromorphic function on $X$ (unique if $U$ is dense). As a corollary, we prove that if $\mathscr X$ is a reduced scheme locally of finite type over an affinoid algebra, every analytic function on $\mathscr X^{\textrm{an}}$ with algebraic zero-locus is algebraic.