Linear sparse differential resultant formulas

TL;DR

This paper introduces determinant-based formulas for eliminating variables in linear differential systems, enabling computation of differential resultants, especially for sparse generic systems, with potential applications in symbolic computation.

Contribution

It presents new determinant formulas for differential elimination in linear systems and connects them to the differential resultant for sparse generic systems.

Findings

Formulas are determinants of coefficient matrices of derivatives.

The formula res(P) has no zero columns if P is 'super essential'.

Applicable to computing differential resultants for sparse systems.

Abstract

Let $\cP$ be a system of $n$ linear nonhomogeneous ordinary differential polynomials in a set $U$ of $n-1$ differential indeterminates. Differential resultant formulas are presented to eliminate the differential indeterminates in $U$ from $\cP$. These formulas are determinants of coefficient matrices of appropriate sets of derivatives of the differential polynomials in $\cP$, or in a linear perturbation $\cP_{\varepsilon}$ of $\cP$. In particular, the formula $\dfres(\cP)$ is the determinant of a matrix $\cM(\cP)$ having no zero columns if the system $\cP$ is "super essential". As an application, if the system $\frak{P}$ is sparse generic, such formulas can be used to compute the differential resultant $\dres(\frak{P})$ introduced by Li, Gao and Yuan in (Proceedings of the ISSAC'2011).