Point counting and Wilkie’s conjecture for non-Archimedean Pfaffian and Noetherian functions

We consider the problem of counting polynomial curves on analytic or definable subsets over the field ${\mathbb{C}}(\!(t)\!)$, as a function of the degree $r$. A result of this type could be expected by analogy with the classical Pila-Wilkie counting theorem in the archimean situation. Some non-archimedean analogs of this type have been developed in the work of Cluckers-Comte-Loeser for the field ${\mathbb{Q}}_p$, but the situation in ${\mathbb{C}}(\!(t)\!)$ appears to be significantly different. We prove that the set of polynomial curves of a fixed degree $r$ on the transcendental part of a subanalytic set over ${\mathbb{C}}(\!(t)\!)$ is automatically finite, but give examples showing that their number may grow arbitrarily quickly even for analytic sets. Thus no analog of the Pila-Wilkie theorem can be expected to hold for general analytic sets. On the other hand we show that if one restricts to varieties defined by Pfaffian or Noetherian functions, then the number grows at most polynomially in $r$, thus showing that the analog of Wilkie's conjecture does hold in this context.