GEOMETRIC PROOFS OF THEOREMS OF AX-KOCHEN AND ERŠOV

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 yie...

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Bibliographic Details
Published in:American journal of mathematics 2016-02, Vol.138 (1), p.181-199
Main Author: Denef, Jan
Format: Article
Language:English
Subjects:
Algebra
Geometry
Theorems
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Summary: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-Eršov transfer principle for local fields, and of quantifier elimination theorems of Basarab and Pas.
ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.2016.0008