Polytopes and simplexes in p-adic fields

by L. Darnière

Annals of Pure and Applied Logic 168 (2016), no. 6, 1284-1307

Submitted on March 22, 2016.

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.

Mathematics Subject Classification

03C64 Model theory of ordered structures; o-minimality
12L12 Model theory [See also 03C60]
12J12 Formally p-adic fields

Electronic version of the paper

Version February 2016 (25 pages) pdf