Linear sparse differential resultant formulas

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).