  Let $K$ be a local field and $f(x)\in K[x]$ be a non-constant polynomial.
When ${\rm char}K=0$, Igusa showed the local zeta function is a rational
function. However, when ${\rm char}K>0$, the rationality of the local zeta
function is unknown in general. In this paper, we study the local zeta
functions for the so-called hybrid polynomials in three variables with
coefficients in a non-archimedean local field of positive characteristic. These
hybrid polynomials were first introduced by Hauser in 2003 to study the
resolution of singularities in positive characteristic. We establish the
rationality theorem for these local zeta functions and list explicitly all the
candidate poles. Our result generalizes the work of Le$\acute{o}$n-Cardenal,
Ibadula and Segers and that of Yin and Hong.

Number Theory