A subdivision-based algorithm for the sparse resultant

Multivariate resultants generalize the Sylvester resultant of two 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 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 resultan...

• Type: Journal
• Year: 2000
• Where: JACM
• Authors: John F. Canny, Ioannis Z. Emiris
• Keywords: Ective Sparse Nullstellensatz, JACM 2000, Mixed Polyhedral Subdivision, Polynomial