Matrix Formulae of Differential Resultant for First Order Generic Ordinary Differential Polynomials

In this paper, a matrix representation for the differential resultant of two generic ordinary differential polynomials \(f_1\) and \(f_2\) in the differential indeterminate \(y\) with order one and arbitrary degree is given. That is, a nonsingular matrix is constructed such that its determinant contains the differential resultant as a factor. Furthermore, the algebraic sparse resultant of \(f_1, f_2, {\updelta } f_1\) and \({\updelta } f_2\) treated as polynomials in \(y, y^{\prime }, y^{\prime \prime }\) is shown to be a nonzero multiple of the differential resultant of \(f_1\) and \(f_2\). Although very special, this seems to be the first matrix representation for a class of nonlinear generic differential polynomials.