Efficient Incremental Algorithms for the Sparse Resultant and the Mixed Volume

Abstract

We propose a new and e cient algorithm for computing the sparse resultant of a system of n + 1 polynomial equations in n unknowns. This algorithm produces a matrix whose entries are coe cients of the given polynomials and is typically smaller than the matrices obtained by previous approaches. The matrix determinant is a nontrivial multiple of the sparse resultant from which the sparse resultant itself can be recovered. The algorithm is incremental in the sense that successively larger matrices are constructed until one is found with the above properties. For multigraded systems, the new algorithm produces optimal matrices, i.e., expresses the sparse resultant as a single determinant. An implementation of the algorithm is described and experimental results are presented. In addition, we propose an e cient algorithm for computing the mixed volume of n polynomials in n variables. This computation provides an upper bound on the number of common isolated roots. A publicly available implementation of the algorithm is presented and empirical results are reported which suggest that it is the fastest mixed volume code to date.

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].…”

“…There exist very efficient algorithms to compute algebraic sparse resultants [14][15][16], which are based on matrix representations for the resultant. How to apply the principles behind these algorithms to compute sparse differential resultants is an important problem.…”

“…The order and degree bounds serve as the termination condition. Precisely, we have computational complexity is single exponential [15,45]. This seems to be the first algorithm which eliminates several variables from nonlinear differential polynomials with a single exponential complexity.…”

“…Sturmfels further introduced the general mixed sparse resultant and gave a single exponential algorithm to compute the sparse resultant [45,46]. 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 [14,15]. D'Andrea further proved that the sparse resultant is the quotient of two Macaulay style determinants [11].…”

“…In recent years, the multivariate resultant has emerged as one of the most powerful computational tools in elimination theory due to its ability to eliminate several variables simultaneously without introducing many extraneous solutions. Many algorithms with best complexity bounds for problems such as polynomial equation solving and first-order quantifier elimination are strongly based on the multivariate resultant [4,5,15,16,26,38].…”