Cell Decomposition for semibounded p-adic sets

2012/05/18 by Eva Leenknegt, Leenknegt, Eva
Computer Science · Mathematics · #03c07 #03c10 #03c64 #11u09 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Logic (math.LO) #Polynomial and algebraic computation #math.LO #msc:03c07 #msc:03c10 #msc:03c64 #msc:11u09

paper · pdf · doi:10.48550/arxiv.1205.4178

20 pages

arxiv created 2012/05/18 · openalex publication_date 2012/05/18 · arxiv updated 2012/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a reduct L∗ 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∗-definable subsets of K coincide with the semi-algebraic subsets of K. Hence structures (K,L∗) 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.

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