Computation of Differential Chow Forms for Prime Differential Ideals

TL;DR

This paper introduces algorithms for computing differential Chow forms of prime differential ideals, establishing bounds on their order and degree, and confirming the Jacobi bound conjecture in this context.

Contribution

It presents new algorithms for differential Chow form computation based on order and degree bounds, validating the Jacobi bound conjecture for these ideals.

Findings

Order bound for prime differential ideals established

Degree bound for differential Chow forms provided

Algorithms with single exponential complexity in key parameters

Abstract

In this paper, we propose algorithms to compute differential Chow forms for prime differential ideals which are given by their characteristic sets. The main algorithm is based on an optimal bound for the order of a prime differential ideal in terms of its characteristic set under an arbitrary ranking, which shows the Jacobi bound conjecture holds in this case. Apart from the order bound, we also give a degree bound for the differential Chow form. In addition, for prime differential ideals given by their characteristic sets under an orderly ranking, a much more simpler algorithm is given to compute its differential Chow form. The computational complexity of both is single exponential in terms of the Jacobi number, the maximal degree of the differential polynomials in the characteristic set and the number of variables.