Differential elimination by differential specialization of Sylvester style matrices

Differential resultant formulas are defined for a system PP of n   ordinary Laurent differential polynomials in n−1n−1 differential variables. These are determinants of coefficient matrices of an extended system of polynomials obtained from PP through derivations and multiplications by Laurent monomials. To start, through derivations, a system ps(P)ps(P) of L   polynomials in L−1L−1 algebraic variables is obtained, which is nonsparse in the order of derivation, as defined in this paper. This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in ps(P)ps(P), to obtain polynomials in the differential elimination ideal generated by PP. If the system satisfies certain conditions, then the formulas obtained are multiples of sparse differential resultants, defined by Li, Yuan and Gao, and provide order and degree bounds in terms of mixed volumes in the generic case.