Polytopes and simplexes in p-adic fields

Luck Darnière (1)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

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.

Mots clés: p-adic simplexes; monotopic division; p-adic polytopes; monohedral division

Domaines: Logique [math.LO]

Luck Darnière. Polytopes and simplexes in p-adic fields. Annals of Pure and Applied Logic, 2016, 168 (6), pp.1284-1307. ⟨hal-01276748v2⟩