In this paper, we first introduce the concept of Laurent differentially essential systems and give a criterion for Laurent differentially essential systems in terms of their supports. Then the sparse differential resultant for a Laurent differentially essential system is defined and its basic properties are proved. In particular, order and degree bounds for the sparse differential resultant are given. Based on these bounds, an algorithm to compute the sparse differential resultant is proposed, which is single exponential in terms of the number of indeterminates, the Jacobi number of the system, and the size of the system.
The multivariate resultant, which gives conditions for an over-determined system of polynomial equations to have common solutions, is a basic concept in algebraic geometry [12,19,23,26,27,41,49]. In recent years, the multivariate resultant is emerged as one of the most powerful computational tools in elimination theory due to its ability to eliminate several variables simultaneously without introducing much extraneous solutions. Many algorithms with best complexity bounds for problems such as polynomial equation solving and first order quantifier elimination, are based on the multivariate resultant [4,5,14,15,42].
In the theory of multivariate resultants, polynomials are assumed to contain all the monomials with degrees up to a given bound. In practical problems, most polynomials are sparse in that they only contain certain fixed monomials. For such sparse polynomials, the multivariate resultant often becomes identically zero and cannot provide any useful information.
As a major advance in algebraic geometry and elimination theory, the concept of sparse resultant was introduced by Gelfand, Kapranov, Sturmfels, and Zelevinsky [19,49]. The degree of the sparse resultant is the Bernstein-Kushnirenko-Khovanskii (BKK) bound [2] instead of the Beźout bound [19,40,50], which makes the computation of the sparse resultant more efficient. The concept of sparse resultant is originated from the work of Gelfand, Kapranov, and Zelevinsky on generalized hypergeometric functions, where the central concept of A-discriminant is studied [17]. Kapranov, Sturmfels, and Zelevinsky introduced the concept of A-resultant [28]. Sturmfels further introduced the general mixed sparse resultant and gave a single exponential algorithm to compute the sparse resultant [49,50]. Canny and Emiris showed that the sparse resultant is a factor of the determinant of a Macaulay style matrix and gave an efficient algorithm to compute the sparse resultant based on this matrix representation [13,14]. D’Andrea further proved that the sparse resultant is the quotient of two Macaulay style determinants similar to the multivariate resultant [11].
Using the analogue between ordinary differential operators and univariate polynomials, the differential resultant for two linear ordinary differential operators was implicitly given by Ore [39] and then studied by Berkovich and Tsirulik [1] using Sylvester style matrices. The subresultant theory was first studied by Chardin [7] for two differential operators and then by Li [38] and Hong [24] for the more general Ore polynomials.
For nonlinear differential polynomials, the differential resultant is more difficult to define and study. The differential resultant for two nonlinear differential polynomials in one variable was defined by Ritt in [44, p.47]. In [55, p.46], Zwillinger proposed to define the differential resultant of two differential polynomials as the determinant of a matrix following the idea of algebraic multivariate resultants, but did not give details. General differential resultants were defined by Carrà-Ferro using Macaulay’s definition of algebraic resultants [6]. But, the treatment in [6] is not complete. For instance, the differential resultant for two generic differential polynomials with positive orders and degrees greater than one is always identically zero if using the definition in [6]. In [54], Yang, Zeng, and Zhang used the idea of algebraic Dixon resultant to compute the differential resultant. 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 [46,47]. In [16], a rigorous definition for the differential resultant of n + 1 differential polynomials in n variables was first presented and its properties were proved. A generic differential polynomial with order o and degree d contains an exponential number of differential monomials in terms of o and d. Thus it is meaningful to study the sparse differential resultant which is the main focus of this paper.
Our first observation is that the sparse differential resultant is related with the nonpolynomial solutions of algebraic differential equations, that is, solutions with non-vanishing derivatives to any order. As a consequence, the sparse differential resultant should be more naturally defined for Laurent differential polynomials. This is similar to the algebraic sparse resultant [19,50], where non-zero solutions of Laurent polynomials are considered.
Consider n + 1 Laurent differential polynomials in n differential indeterminates Y = {y 1 , . . . , y n }:
where u ik ∈ E are differentially independent over Q and M ik are Laurent differential monomials in Y. As explained later in this paper, we can assume that M ik are monomials with nonnegative exponent vectors α ik . Let s i = ord(P i , Y) and denote M ik /M i0 = n j=1 s i l=0 (y
, where y (l) j
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