Uniform parameterization of subanalytic sets and diophantine applications

TL;DR

This paper develops new uniform parameterization theorems for subanalytic sets, introduces analytic quasi-parameterization to achieve uniformity, and applies these results to improve bounds on rational points in diophantine geometry.

Contribution

It introduces uniform $C^r$ and analytic quasi-parameterization theorems for subanalytic sets, advancing diophantine applications and rational point counting.

Findings

Established polynomial bounds for $C^r$-parameterizations.

Introduced analytic quasi-parameterization for uniform results.

Achieved $( ext{log } H)^{O(1)}$ bounds on rational points.

Abstract

We prove new parameterization theorems for sets definable in the structure $\mathbb{R}_{an}$ (i.e. for globally subanalytic sets) which are uniform for definable families of such sets. We treat both $C^r$-parameterization and (mild) analytic parameterization. In the former case we establish a polynomial (in $r$) bound (depending only on the given family) for the number of parameterizing functions. However, since uniformity is impossible in the latter case (as was shown by Yomdin via a very simple family of algebraic sets), we introduce a new notion, analytic quasi-parameterization (where many-valued complex analytic functions are used), which allows us to recover a uniform result. We then give some diophantine applications motivated by the question as to whether the $H^{o(1)}$ bound in the Pila-Wilkie counting theorem can be improved, at least for certain reducts of $\mathbb{R}_{an}$. Both parameterization results are shown to give uniform $(\log H)^{O(1)}$ bounds for the number of rational points of height at most $H$ on $\mathbb{R}_{an}$-definable Pfaffian surfaces. The quasi-parameterization technique produces the sharper result, but the uniform $C^r$-parametrization theorem has the advantage of also applying to $\mathbb{R}_{an}^{pow}$-definable families.