Polytopes and simplexes in p-adic fields

TL;DR

This paper develops a topological framework for polytopes and simplexes in p-adic fields, aiming to establish a triangulation theory analogous to that in real algebraic geometry.

Contribution

It introduces the notions of polytopes and simplexes in p-adic fields and constructs a method to subdivide polytopes into simplexes with specified faces and shapes.

Findings

Faces of p-adic polytopes form a rooted tree under specialization.

Simplexes are characterized by their face tree being a chain.

A construction method for subdividing polytopes into simplexes is provided.

Abstract

We introduce topological notions of polytopes and simplexes, the latter being expected to play in p-adically closed fields the role played by real simplexes in the classical results of triangulation of semi-algebraic sets over real closed fields. We prove that the faces of every p-adic polytope are polytopes and that they form a rooted tree with respect to specialisation. Simplexes are then defined as polytopes whose faces tree is a chain. Our main result is a construction allowing to divide every p-adic polytope in a complex of p-adic simplexes with prescribed faces and shapes.