Uniform parameterization of subanalytic sets and diophantine applications

We prove new parameterization theorems for sets definable in the structure ℝan (i.e. for globally subanalytic sets) which are uniform for definable families of such sets. We treat both Cr-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 Ho(1) bound in the Pila-Wilkie counting theorem can be improved, at least for certain reducts of Ran. Both parameterization results are shown to give uniform (logH)O(1) bounds for the number of rational points of height at most H on ℝan-definable Pfaffian surfaces. The quasi-parameterization technique produces the sharper result, but the uniform Cr-parameterization theorem has the advantage of also applying to ℝpow/an -definable families.