## 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