Linear sparse differential resultant formulas

Abstract

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 P ε of P. In particular, the formula ∂FRes(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 P is sparse generic, such formulas can be used to compute the differential resultant ∂Res(P) introduced by Li, Gao and Yuan in [20].

Sparse Differential Resultant for Laurent Differential Polynomials

“…It is useful to represent the sparse differential resultant as the quotient of two determinants, as done in [11,15] in the algebraic case. In the differential case, we do not have such formulas, even in the simplest case of the resultant for two generic differential polynomials in one variable [49] or a system of linear sparse differential polynomials [43]. In [43], for a sparse linear differential system S, Rueda gave an enlarged system S 1 of S such that S 1 has a matrix representation and the sparse differential resultant of S can be obtained from the determinant of S 1 .…”

“…Although efficient, this approach is not complete, because it is not proved that the differential resultant can always be computed in this way. Differential resultants for linear ordinary differential polynomials were studied by Rueda-Sendra [43,44]. In [17], a rigorous definition for the differential resultant of n + 1 differential polynomials in n variables was first presented and its properties were proved.…”

Index reduction of differential algebraic equations by differential Dixon resultant

“…The underlying theory is the differential algebra of Ritt [28] and Kolchin [15]. Differential elimination algorithm in algebraic elimination theory is an active field and powerful tools with many important applications [7,10,26,29,34,35]. Almost all of the authors focus on the differential elimination theory for ODEs.…”

Sparse difference resultant

“…Secondly, the criterion for Laurent transformally essential systems given in Section 3.3 is quite different and much simpler than its differential counterpart given in [27]. Also, determinant representations for the sparse difference resultant and the difference resultant are given in Section 5 and Section 7, but such a representation is still not known for differential resultants [38,30,31]. Finally, there does not exist a definition for homogeneous difference polynomials, and the definition we give in this paper is different from its differential counterpart [25].…”