An efficient algorithm for the sparse mixed resultant

Abstract

Abstract. We propose a compact formula for the mixed resultant o f a system of n+1 sparse Laurent polynomials in n variables. Our approach i s conceptually simple and geometric, in that it applies a mixed subdivision to the Minkowski Sum of the input Newton polytopes. It constructs a matrix whose determinant is a non-zero multiple of the resultant so that the latter can be de ned as the GCD of n + 1 such determinants. For any specialization of the coe cients there are two methods which use one extra perturbation variable and return the resultant. Our algorithm is the rst to present a determinantal formula for arbitrary systems; moreover, its complexity for unmixed systems is polynomial in the resultant degree. Further empirical results suggest that this is the most e cient method to date for sparse elimination.

The Computational Complexity of the Chow Form

“…Several effective procedures were proposed to compute it (see e.g. [50], [9], [10]). Recently, C. D'Andrea has obtained an explicit determinantal formula which extends Macaulay's formula to the sparse case ( [15]).…”

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“…Canny and Emiris [5] applied these coarse decompositions to give an efficient algorithm for computing the sparse mixed resultant. More precisely, for each coarse TCMD A w they constructed a square matrix M u of size roughly card(Qn Z n ) and having entries c i,a and 0, whose determinant is a nonzero multiple of K. A key point of their construction is that the extreme term init u (R) appears on the main diagonal of the matrix M u .…”

“…Nineteen of these lie in the mixed cells of A w + 6. The four remaining points are (2,5), (2,6), (1,3), (1,4). If we order the set £ such that these four extraneous points come first, then our matrix has the structure where I 4 , denotes the 4 x 4-unit matrix and 0 denotes the 19 x 4-zero matrix.…”

“…Canny and Emiris [5] recently gave an efficient algorithm, based on a determinantal formula, for computing the sparse mixed resultant. In Section 3 we generalize the Canny-Emiris formula by showing that for each extreme monomial m of R there exists a determinant as in [5], for which m appears as a factor of the main diagonal product.…”

“…In Section 3 we generalize the Canny-Emiris formula by showing that for each extreme monomial m of R there exists a determinant as in [5], for which m appears as a factor of the main diagonal product.…”