Effective Pila--Wilkie bounds for unrestricted Pfaffian surfaces

TL;DR

This paper establishes effective bounds on the number of rational points of bounded height on certain Pfaffian surfaces, including graphs of unrestricted Pfaffian functions, advancing the understanding of rational points in these geometric contexts.

Contribution

It provides effective Pila--Wilkie estimates for a broad class of Pfaffian surfaces, including those defined by unrestricted Pfaffian functions, which was not previously achieved.

Findings

Established bounds for rational points on Pfaffian surfaces

Included graphs of unrestricted Pfaffian functions in the results

Enhanced understanding of rational points in Pfaffian geometry

Abstract

We prove effective Pila--Wilkie estimates for the number of rational points of bounded height lying on certain surfaces defined by Pfaffian functions. The class of surfaces to which our result applies includes, for instance, graphs of unrestricted Pfaffian functions defined on the plane.