Cell Decomposition for semibounded p-adic sets

TL;DR

This paper introduces a cell decomposition theorem for a reduct of p-adic fields with restricted multiplication, enabling quantifier elimination and a detailed understanding of definable functions in this setting.

Contribution

It establishes cell decomposition, quantifier elimination, and characterizes definable functions in a reduct of p-adic fields where multiplication is restricted to neighborhoods of zero.

Findings

Cell decomposition holds for the reduct of p-adic fields.

Quantifier elimination is achieved in this language.

Multi-plication is only definable on bounded sets.

Abstract

We study a reduct L\ast of the ring language where multiplication is restricted to a neighbourhood of zero. The language is chosen such that for p-adically closed fields K, the L\ast-definable subsets of K coincide with the semi-algebraic subsets of K. Hence structures (K,L\ast) can be seen as the p-adic counterpart of the o-minimal structure of semibounded sets. We show that in this language, p-adically closed fields admit cell decomposition, using cells similar to p-adic semi-algebraic cells. From this we can derive quantifier-elimination, and give a characterization of definable functions. In particular, we conclude that multi- plication can only be defined on bounded sets, and we consider the existence of definable Skolem functions.