1,240 research outputs found
Igusa's p-adic local zeta function associated to a polynomial mapping and a polynomial integration measure
For p prime, we give an explicit formula for Igusa's local zeta function
associated to a polynomial mapping f=(f_1,...,f_t): Q_p^n -> Q_p^t, with
f_1,...,f_t in Z_p[x_1,...,x_n], and an integration measure on Z_p^n of the
form |g(x)||dx|, with g another polynomial in Z_p[x_1,...,x_n]. We treat the
special cases of a single polynomial and a monomial ideal separately. The
formula is in terms of Newton polyhedra and will be valid for f and g
sufficiently non-degenerated over F_p with respect to their Newton polyhedra.
The formula is based on, and is a generalization of results of Denef -
Hoornaert, Howald et al., and Veys - Zuniga-Galindo.Comment: 20 pages, 5 figures, 2 table
Definable sets, motives and p-adic integrals
We associate canonical virtual motives to definable sets over a field of
characteristic zero. We use this construction to show that very general p-adic
integrals are canonically interpolated by motivic ones.Comment: 45 page
Trees of definable sets over the p-adics
To a definable subset of Z_p^n (or to a scheme of finite type over Z_p) one
can associate a tree in a natural way. It is known that the corresponding
Poincare series P(X) = \sum_i N_i X^i is rational, where N_i is the number of
nodes of the tree at depth i. This suggests that the trees themselves are far
from arbitrary. We state a conjectural, purely combinatorial description of the
class of possible trees and provide some evidence for it. We verify that any
tree in our class indeed arises from a definable set, and we prove that the
tree of a definable set (or of a scheme) lies in our class in three special
cases: under weak smoothness assumptions, for definable subsets of Z_p^2, and
for one-dimensional sets.Comment: 33 pages, 1 figur
Motivic integration and the Grothendieck group of pseudo-finite fields
We survey our recent work on an extension of the theory of motivic
integration, called arithmetic motivic integration. We developed this theory to
understand how p-adic integrals of a very general type depend on p.Comment: 11 page
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