论文标题
点计数和Wilkie对非Archimedean Pfaffian和Noetherian功能的猜想
Point counting and Wilkie's conjecture for non-archimedean Pfaffian and Noetherian functions
论文作者
论文摘要
我们考虑在分析或可定义的子集上计数多项式曲线的问题$ {\ mathbb {c}}}}（\！（t）\！）$，作为度$ r $的函数。与经典的pila-wilkie计算定理的经典情况下，这种类型的结果可以预期。 这种类型的某些非架构类似物是在cluckers-comte-loeser的作品中开发的，用于字段$ {\ mathbb {q}} _ p $，但是$ {\ mathbb {c}}}（\！（t）\！）$中的情况似乎是显着不同的。我们证明，在$ {\ mathbb {c}}}（\！（t）\！）$上的固定度$ r $ $ r $的一组固定度$ r $的多项式曲线是自动有限的，但给出了示例，显示其数量可能快速地增长，即使分析集更快地增长。因此，对于一般的分析集，没有预期pila-wilkie定理的类似物。另一方面，我们表明，如果一个人限制了由Pfaffian或Noetherian函数定义的品种，那么该数字在$ r $中最多地增长，因此表明Wilkie猜想的模拟确实存在于这种情况下。
We consider the problem of counting polynomial curves on analytic or definable subsets over the field ${\mathbb{C}}(\!(t)\!)$, as a function of the degree $r$. A result of this type could be expected by analogy with the classical Pila-Wilkie counting theorem in the archimean situation. Some non-archimedean analogs of this type have been developed in the work of Cluckers-Comte-Loeser for the field ${\mathbb{Q}}_p$, but the situation in ${\mathbb{C}}(\!(t)\!)$ appears to be significantly different. We prove that the set of polynomial curves of a fixed degree $r$ on the transcendental part of a subanalytic set over ${\mathbb{C}}(\!(t)\!)$ is automatically finite, but give examples showing that their number may grow arbitrarily quickly even for analytic sets. Thus no analog of the Pila-Wilkie theorem can be expected to hold for general analytic sets. On the other hand we show that if one restricts to varieties defined by Pfaffian or Noetherian functions, then the number grows at most polynomially in $r$, thus showing that the analog of Wilkie's conjecture does hold in this context.
