亨泽尔极小曲线的皮拉 - 威尔基定理

Point counting and Wilkie's conjecture for non-archimedean Pfaffian and Noetherian functions

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 s…

Wilkie's conjecture for restricted elementary functions

We consider the structure ${\mathbb R}^{\mathrm{RE}}$ obtained from $({\mathbb R},<,+,\cdot)$ by adjoining the restricted exponential and sine functions. We prove Wilkie's conjecture for sets definable in this structure: the number of rational points of height $H$ in the transcendental part of any definable set is bounded by a polynomial in $\log H$. We also prove two refined conjectures due to Pila concerning the density of algebraic points from a fixed number field, or with a fixed algebraic degree, for ${\mathbb R}^{\mathrm{RE}}$-definable sets.

T-模与皮拉-威尔基估计

T-modules and Pila-Wilkie estimates

该研究针对正特征函数域算术里的特殊T-模，给出其对应解析集超越部分有理点个数的上界估计，为证明这类T-模的马宁-芒福德猜想奠定了基础。

Hensel minimality I

We present a framework for tame geometry on Henselian valued fields which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show existence of t-stratifications in Hensel minimal structures and Taylor approximation results which are key to non-archimedean versions of Pila-Wilkie point counting, Yomdin's parameterization results and to motivic integration. In this firs…

Applications of the hyperbolic Ax-Schanuel conjecture

In 2014, Pila and Tsimerman gave a proof of the Ax-Schanuel conjecture for the $j$-function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax-Schanuel conjecture. In this article, we show that the hyperbolic Ax-Schanuel conjecture can be used to reduce the Zilber-Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila-Wilkie counting theorem and several arithmetic conjectures. Our…

A note on o-minimal flows and the Ax--Lindemann--Weierstrass theorem for abelian varieties over $\mathbb{C}$

In this short note we present an elementary proof of the Ax-Lindemann-Weierstrass theorem for abelian and semi-abelian varieties. The proof uses ideas of Pila, Ullmo, Yafaev, Zannier and is based on basic properties of sets definable in o-minimal structures. It does not use the Pila-Wilkie counting theorem.

用于Drinfeld模和Drinfeld模曲线的Pila-Zannier策略

The Pila-Zannier strategy for Drinfeld modules and Drinfeld modular curves

研究将Pila-Zannier策略扩展到Drinfeld模，证明了相关定理类似物。在特征零中，用刚性解析Pila-Wilkie定理替代基于\(o\)-极小性的计数步骤，通过点数论证建立函数超越性输入，实现该策略在函数域的应用。

尖锐 o-极小结构中的对数-指数准备

Logarithmic--exponential preparation in sharply o-minimal structures

本文通过复解析方法，为实数无限制指数结构建立了尖锐 o-极小性、Wilkie 定理及猜想的尖锐形式，并推广了有理点插值定理。

Hensel最小性，p进指数和Tate统一化

Hensel minimality, $p$-adic exponentiation and Tate uniformization

本文利用Hensel最小性研究p进指数和Tate统一化在非阿基米德设定下的性质，证明其在Hensel最小结构中的存在，并探讨了p进Schanuel猜想的统一版本及椭圆曲线交集的密度问题。

赋值域中超越曲线上有理点的计数

Counting rational points on transcendental curves in valued fields

本文在任意1-h-极小域中建立超越曲线上有理点数的上界，将Pila–Wilkie型计数定理从p-进域推广到混合特征赋值域，方法上避免r次幂映射并采用参数化与行列式方法。

整数值 o-极小函数

Integer-valued o-minimal functions

本文研究在大整数处取整数值的 $\mathbb{R}_{\textrm{an},\exp}$-可定义函数，证明在接近 $2^x$ 的增长界下其最终为多项式与 $P_2(x)2^x$ 之和，并将 Wilkie 定理推广到 $k$-concordant 函数。