Difference Chow Form

In this paper, the generic intersection theory for difference varieties is presented. Precisely, the intersection of an irreducible difference variety of dimension $d > 0$ and order $h$ with a generic difference hypersurface of order $s$ is shown to be an irreducible difference variety of dimension $d-1$ and order $h+s$. Based on the intersection theory, the difference Chow form for an irreducible difference variety is defined. Furthermore, it is shown that the difference Chow form of an irreducible difference variety $V$ is transformally homogenous and has the same order as $V$.