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 field C((t)) as a function degree r. A result this type could be expected by analogy with classical Pila–Wilkie theorem in Archimedean situation. Some non-Archimedean analogues have been developed work Cluckers, Comte, and Loeser for Qp, but situation appears to significantly different. prove that set fixed r transcendental part subanalytic is automatically finite, we give examples show their number may grow arbitrarily quickly even sets. Thus no analogue can hold general On other hand, if one restricts varieties defined Pfaffian Noetherian functions, then grows at most polynomially r, thus showing Wilkie conjecture does context.