Geometric proofs of theorems of Ax-Kochen and Ersov

TL;DR

This paper presents an algebraic geometric proof of the Ax-Kochen theorem on p-adic equations, avoiding logic-based methods and introducing weak toroidalization, leading to new proofs of related transfer principles and quantifier elimination results.

Contribution

It provides a novel geometric proof of the Ax-Kochen theorem that does not rely on mathematical logic, using weak toroidalization of morphisms.

Findings

Geometric proof of Ax-Kochen theorem without logic

New proofs of Ax-Kochen-Ersov transfer principle

Alternative proof of quantifier elimination theorems

Abstract

We give an algebraic geometric proof of the Theorem of Ax and Kochen on p-adic diophantine equations in many variables. Unlike Ax-Kochen's proof, ours does not use any notions from mathematical logic and is based on weak toroidalization of morphisms. We also show how this geometric approach yields new proofs of the Ax-Kochen-Ersov transfer principle for local fields, and of quantifier elimination theorems of Basarab and Pas.