Igusa's p-adic local zeta function associated to a polynomial mapping and a polynomial integration measure

Details Igusa's p-adic local zeta function associated to a polynomial mapping and a polynomial integration measure Igusa's p-adic local zeta function associated to a polynomial mapping and a polynomial integration measure

2010-12-22

arxiv.org/abs/1012.5087v2

Mathematics

Number Theory

20 pages, 5 figures

Scientific paper

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.

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