Abstract
We propose a compact formula for the mixed resultant of a system of n+1 sparse Laurent polynomials in n variables. Our approach is 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 defined as the GCD of n + 1 such determinants. For any specialization of the coefficients there are two methods which use one extra perturbation variable and return the resultant. Our algorithm is the first 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 efficient method to date for sparse elimination.
Supported by a David and Lucile Packard Foundation Fellowship and by NSF Presidential Young Investigator Grant IRI-8958577.
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Canny, J., Emiris, I. (1993). An efficient algorithm for the sparse mixed resultant. In: Cohen, G., Mora, T., Moreno, O. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1993. Lecture Notes in Computer Science, vol 673. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56686-4_36
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DOI: https://doi.org/10.1007/3-540-56686-4_36
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