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

Abstract

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 Archimedean situation.

Some non-Archimedean analogues of this type have been developed in the work of Cluckers, Comte, and 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 we give examples that show their number may grow arbitrarily quickly even for analytic sets. Thus no analogue 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 analogue of the Wilkie conjecture does hold in this context.