## 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

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 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