Definable functions in tame expansions of algebraically closed valued fields

TL;DR

This paper investigates the structure of definable functions in tame expansions of algebraically closed valued fields, establishing factorizations and boundedness results, with implications for the behavior of functions and their domains.

Contribution

It provides new factorizations of definable functions over the value group and residue structure, demonstrating polynomial boundedness in certain tame expansions.

Findings

Definable functions can be factorized over the value group.

Tame expansions with value group  are polynomially bounded.

Domains of definable functions can be partitioned into finite, locally constant, and Jacobian-satisfying parts.

Abstract

In this article we study definable functions in tame expansions of algebraically closed valued fields. For a given definable function we have two types of results: of type (I), which hold at a neighborhood of infinity, and of type (II), which hold locally for all but finitely many points in the domain of the function. In the first part of the article, we show type (I) and (II) results concerning factorizations of definable functions over the value group. As an application, we show that tame expansions of algebraically closed valued fields having value group $\mathbb{Q}$ (like $\mathbb{C}_p$ and $\overline{\mathbb{F}_p}^{alg}(\!(t^\mathbb{Q})\!)$) are polynomially bounded. In the second part, under an additional assumption on the asymptotic behavior of unary definable functions of the value group, we extend these factorizations over the residue multiplicative structure $\mathrm{RV}$. In characteristic 0, we obtain as a corollary that the domain of a definable function $f\colon X\subseteq K\to K$ can be partitioned into sets $F\cup E\cup J$, where $F$ is finite, $f|E$ is locally constant and $f|J$ satisfies locally the Jacobian property.