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

Pas, Johan

J. Reine Angew. Math. 399, 137-172 (1989).

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)

Keywords:

cell decomposition theorem for p-adic fields; uniform quantifier elimination; Igusa local zeta function