$b$-minimality

从胞腔分解到动机积分、Hensel 极小性与点计数

From Cell Decomposition to Motivic Integration, Hensel Minimality, and Point Counting

本文回顾 Denef 胞腔分解在动机积分与 Hensel 极小性中的催化作用，及其在 Langlands 纲领和可定义集有理点研究中的应用，并讨论相关开放问题。

Clustered Cell Decomposition in P-Minimal Structures

We prove that in a $P$-minimal structure, every definable set can be partitioned as a finite union of classical cells and regular clustered cells. This is a generalization of previously known cell decomposition results by Denef and Mourgues, which were dependent on the existence of definable Skolem functions. Clustered cells have the same geometric structure as classical, Denef-type cells, but do not have a definable function as center. Instead, the center is given by a definable set whose fibers are finite unions of balls.

Cell decomposition and classification of definable sets in p-optimal fields

We prove that for p-optimal fields (a very large subclass of p-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef's paper [Invent. Math, 77 (1984)]. We derive from it the existence of definable Skolem functions and strong p-minimality. Then we turn to strongly p-optimal fields satisfying the Extreme Value Property (a property which in particular holds in fields which are elementarily equivalent to a p-adic one). For such fields K, we prove that every definable subset of KxK^d whose fibers are inverse images by the valuation o…