## Uniform $$p$$-adic cell decomposition and local zeta functions.(English)Zbl 0666.12014

We prove a cell decomposition theorem for $$p$$-adic fields, uniform in the prime $$p$$, and give some applications of this theorem.
A first implication is a uniform quantifier elimination for the fields of $$p$$-adic numbers.
As a second application, we reprove results of Denef on the dependence on $$p$$ of the Igusa local zeta function. In this context we also obtain new results on $$p$$-adic integrals over sets definable in a language with cross section.
Reviewer: Johan Pas (Leuven)

### MSC:

 11S40 Zeta functions and $$L$$-functions 03C10 Quantifier elimination, model completeness, and related topics 03C60 Model-theoretic algebra 11U09 Model theory (number-theoretic aspects)
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