Classification of semi-algebraic $p$-adic sets up to semi-algebraic bijection

TL;DR

This paper establishes that infinite p-adic semi-algebraic sets are semi-algebraically bijective if and only if they share the same dimension, providing a complete classification criterion.

Contribution

It proves a classification theorem for infinite p-adic semi-algebraic sets based solely on their dimension, resolving a fundamental question in p-adic semi-algebraic geometry.

Findings

Semi-algebraic sets are classified by dimension up to bijection.

Two infinite p-adic semi-algebraic sets are isomorphic iff they have the same dimension.

The result simplifies understanding of the structure of p-adic semi-algebraic sets.

Abstract

We prove that two infinite p-adic semi-algebraic sets are isomorphic (i.e. there exists a semi-algebraic bijection between them) if and only if they have the same dimension.