A subdivision-based algorithm for the sparse resultant

John F. Canny, Ioannis Z. Emiris
2000 Journal of the ACM  
Multivariate resultants generalize the Sylvester resultant o f t wo polynomials and characterize the solvability of a polynomial system. They also reduce the computation of all common roots to a problem in linear algebra. We propose a determinantal formula for the sparse resultant of an arbitrary system of n + 1 polynomials in n variables. This resultant generalizes the classical one and has signi cantly lower degree for polynomials that are sparse in the sense that their mixed volume is lower
more » ... ed volume is lower than their B zout number. Our algorithm uses a mixed polyhedral subdivision of the Minkowski sum of the Newton polytopes in order to construct a Newton matrix. Its determinant is a nonzero multiple of the sparse resultant and the latter equals the GCD of at most n + 1 such determinants. This construction implies a restricted version of an e ective sparse Nullstellensatz. For an arbitrary specialization of the coe cients there are two methods which use one extra variable and yield the sparse resultant. This is the rst algorithm to handle the general case with complexity polynomial in the resultant degree and simply exponential in n. We conjecture its extension to producing an exact rational expression for the sparse resultant.
doi:10.1145/337244.337247 fatcat:z7z4myffu5a65m7jiug7jaeftu