Uniform parameterization of subanalytic sets and diophantine applications. (arXiv:1605.05916v2 [math.NT] UPDATED)

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 rat 查看全文>>