Difference Chow Form

TL;DR

This paper develops an intersection theory for difference varieties and introduces the difference Chow form, demonstrating its properties such as transformal homogeneity and order preservation.

Contribution

It presents the first intersection theory for difference varieties and defines the difference Chow form with key properties.

Findings

Intersection of difference varieties with hypersurfaces reduces dimension by one.

Difference Chow form is transformally homogeneous.

Order of the difference Chow form equals the original variety's order.

Abstract

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 $V$.