Partial Differential Chow Forms and a Type of Partial Differential Chow varieties

TL;DR

This paper develops an intersection theory for partial differential varieties, introduces the concept of partial differential Chow forms, and proves the existence of corresponding Chow varieties, extending classical differential algebra concepts.

Contribution

It defines partial differential Chow forms for irreducible varieties and establishes their fundamental properties, along with proving the existence of partial differential Chow varieties.

Findings

Established intersection theory with quasi-generic hypersurfaces

Defined and analyzed partial differential Chow forms

Proved existence of partial differential Chow varieties

Abstract

We first present an intersection theory of partial differential varieties with quasi-generic differential hypersurfaces. Then based on the generic intersection theory, we define the partial differential Chow form for an irreducible partial differential variety $V$ of Kolchin polynomial $\omega_V(t)=(d+1){t+m\choose m}-{t+m-s\choose m}$. And we establish for the partial differential Chow form most of the basic properties of the ordinary differential Chow form. Furthermore, we prove the existence of a type of partial differential Chow varieties.