Linear Sparse Differential Resultant Formulas

Let P 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 P. These formulas are determinants of coefficient matrices of appropriate sets of derivatives of the differential polynomials in P, or in a linear perturbation Pe of P. In particular, the formula dfres(P) is the determinant of a matrix M(P) having no zero columns if the system P is ``super essential". As an application, if the system PP is sparse generic, such formulas can be used to compute the differential resultant dres(PP) introduced by Li, Gao and Yuan

Sparse differential resultant formulas: between the linear and the nonlinear case

A matrix representation of the sparse differential resultant is the basis for efficient computation algorithms, whose study promises a great contribution to the development and applicability of differential elimination techniques. It is shown how sparse linear differential resultant formulas provide bounds for the order of derivation, even in the nonlinear case, and they also provide (in many cases) the bridge with results in the nonlinear algebraic case

Differential elimination by differential specialization of Sylvester style matrices

Differential resultant formulas are defined, for a system \cP of $n$ ordinary Laurent differential polynomials in $n-1$ differential variables. These are determinants of coefficient matrices of an extended system of polynomials obtained from \cP through derivations and multiplications by Laurent monomials. To start, through derivations, a system \ps(\cP) of $L$ polynomials in $L-1$ algebraic variables is obtained, which is non sparse in the order of derivation. This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in \ps(\cP), to obtain polynomials in the differential elimination ideal generated by \cP. The formulas obtained are multiples of the sparse differential resultant defined by Li, Yuan and Gao, and provide order and degree bounds in terms of mixed volumes in the generic case

Matrix Formula 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 non-singular matrix is constructed such that its determinant contains the differential resultant as a factor. Furthermore, the algebraic sparse resultant of $f_1, f_2, \delta f_1, \delta f_2$ treated as polynomials in $y, y', y"$ is shown to be a non-zero multiple of the differential resultant of $f_1, f_2$. Although very special, this seems to be the first matrix representation for a class of nonlinear generic differential polynomials

Differential resultants of super essential systems of linear OD-polynomials

The sparse differential resultant dres(P) of an overdetermined system P of generic nonhomogeneous ordinary differential polynomials, was formally defined recently by Li, Gao and Yuan (2011). In this note, a differential resultant formula dfres(P) is defined and proved to be nonzero for linear "super essential" systems. In the linear case, dres(P) is proved to be equal, up to a nonzero constant, to dfres(P*) for the supper essential subsystem P* of P