Hensel 极小性 II：混合特征与丢番图应用
Hensel minimality II: Mixed characteristic and a diophantine application
AI总结：
本文为特征零 Hensel 赋值域建立 Hensel 极小性框架，处理混合特征情形，证明 Jacobian 性质、Taylor 逼近、胞腔分解等几何结果，并应用于计数曲线上有界高度有理点。
摘要：
在本文及前作第一部分 \\(\cite{CHR}\\) 中，我们为特征零的 Hensel 赋值域上的驯顺几何建立了一个框架，称为 Hensel 极小性。它补充了 \\(\cite{CHR}\\) 中关于混合特征情形的处理。Hensel 极小性受到 o-极小性及其在实几何和丢番图应用中的作用的启发。我们为 Hensel 极小结构发展了此前仅在更强或更少公理化假设下已知的几何结果和应用，这些结果通常与 o-极小结构中的结果相对应。我们在 Hensel 极小性（更精确地，\\(1\\)-h-极小性）下证明了 Jacobian 性质、可定义函数的强 Taylor 逼近形式、resplendency 结果和胞腔分解。我们获得了在 Hensel 极小曲线上计数有界高度有理点的丢番图应用。
英文摘要：
In this paper together with the preceding Part I \cite{CHR}, we develop a framework for tame geometry on Henselian valued fields of characteristic zero, called Hensel minimality. It adds to \cite{CHR} the treatment of the mixed characteristic case. Hensel minimality is inspired by o-minimality and its role in real geometry and diophantine applications. We develop geometric results and applications for Hensel minimal structures that were previously known only under stronger or less axiomatic assumptions, and which often have counterparts in o-minimal structures. We prove a Jacobian property, a strong form of Taylor approximations of definable functions, resplendency results and cell decomposition, all under Hensel minimality, more precisely, $1$-h-minimality. We obtain a diophantine application of counting rational points of bounded height on Hensel minimal curves.
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