Sparse Resultant under Vanishing Coefficients

• Authors: Manfred Minimair
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2003

The main question of this paper is: What happens to the sparse (toric) resultant under vanishing coefficients? More precisely, let f1,…,fn be sparse Laurent polynomials with supports \cal A1,…,\cal An and let \tilde{\cal A}1 ⊃ \cal A1. Naturally a question arises: Is the sparse resultant of f1,f2,…,fn with respect to the supports \tilde{\cal A}1,\cal A2,…,\cal An in any way related to the sparse resultant of f1,f2,…,fn with respect to the supports \cal A1,\cal A2,…,\cal An? The main contribution of this paper is to provide an answer. The answer is important for applications with perturbed data where very small coefficients arise as well as when one computes resultants with respect to some fixed supports, not necessarily the supports of the fi's, in order to speed up computations. This work extends some work by Sturmfels on sparse resultant under vanishing coefficients. We also state a corollary on the sparse resultant under powering of variables which generalizes a theorem for Dixon resultant by Kapur and Saxena. We also state a lemma of independent interest generalizing Pedersen's and Sturmfels' Poisson-type product formula.