Poles of the Igusa local zeta function of some hybrid polynomials

Hybrid polynomials were introduced by Hauser [6] in connection with the problem of extending Hinorakaʼs resolution of singularities theorem to fields of positive characteristic. In this paper we study the local zeta function associated to some hybrid polynomials defined over a non-archimedean local field of positive characteristic, by using essentially the π-adic stationary phase formula. We show the rationality of these local zeta functions and we describe completely its poles. For this class of polynomials we also met the classical results about exponential sums mod πm.