A Thom-Sebastiani Theorem in Characteristic p

Let $k$ be a perfect field of characteristic $p$, let $f_i:X_i\to\mathbb A_k^1$ $(i=1,2)$ be two $k$-morphism of finite type, and let $f:X_1\times_k X_2\to \mathbb A_k^1$bethemorphismdefinedby$f(z_1,z_2)=f_1(z_1)+f_2(z_2)$\mathrm{.}Foreach$i\in{1,2}$,let$x_i$bea$k$-rationalpointinthefiber$f_i^{-1}(0)$suchthat$f_i$issmoothon$X_i-{x_i}$\mathrm{.}Usingthe$\ell$-adic Fourier transformation and the stationary phase principle of Laumon, we prove that the vanishing cycle of $f$ at $x=(x_1,x_2)$ is the convolution product of the vanishing cycles of $f_i$ at $x_i$ $(i=1,2)$.