Counting rational points on transcendental curves in valued fields

Abstract

We prove upper bounds on the number of rational points on transcendental curves in arbitrary $1$-h-minimal fields, similar to the Pila--Wilkie counting theorem in the o-minimal setting. These results extend results due to Cluckers--Comte--Loeser from $p$-adic fields to arbitrary valued fields of mixed characteristic. Our methods rely on parametrizations, where we avoid the usage of $r$-th power maps, combined with the determinant method.