Partial differential Chow forms and a type of partial differential Chow varieties

Abstract

We first present an intersection theory of irreducible partial differential varieties with quasi-generic differential hypersurfaces. Then, we define partial differential Chow forms for irreducible partial differential varieties whose Kolchin polynomials are of the form $\omega \left(t\right)=(d+1)\left(\begin{array}{c}t+m\\ m\end{array}\right)-\left(\begin{array}{c}t+m-s\\ m\end{array}\right).$ And we establish for partial differential Chow forms most of the basic properties of their ordinary differential counterparts. Furthermore, we prove that a certain type of partial differential Chow varieties exist.

Communicated by Jason Bell

Keywords:

• Kolchin polynomial
• partial differential Chow form
• partial differential Chow variety
• quasi-generic differential intersection theory

2010 Mathematics Subject Classification:

• 12H05 (primary)
• 03C60

Acknowledgements

The author is grateful to the anonymous referee for the helpful comments and constructive suggestions on a previous version of this manuscript.

Notes

1 By saying η free from $\mathcal{F}〈\mathbb{U}〉,$ we mean that $\mathbb{U}$ is a set of Δ-$\mathcal{F}〈\eta 〉$-indeterminates.

2 Here, the orderly ranking is assumed to guarantee that the order of $h_l$ is bounded by t and thus obtain $H\left(\mathbb{Y}\right)\ne 0,$ when following the proof of Theorem 3.2 to prove the corollary.

Additional information

Funding

This work was supported by NSFC under Grants (11971029, 11688101, 11671014, and 11301519).