Sparse resultant under vanishing coefficients

Minimair, Manfred

Journal of Algebraic Combinatorics, Tome 18 (2003) no. 1, pp. 53-73.

Résumé

Summary: The main question of this paper is: What happens to the sparse $( toric)$ resultant under vanishing coefficients? More precisely, let $f _{1},\dots , f _{n}$ be sparse Laurent polynomials with supports $A _{1},\dots , A _{ n } A_{1} supA _{1}$. Naturally a question arises: Is the sparse resultant of $f _{1}, f _{2},\dots , f _{n}$ with respect to the supports $A _{1}, A _{2},\dots , A _{ n }$ in any way related to the sparse resultant of $f _{1}, f _{2},\dots , f _{n}$ with respect to the supports $A _{1}, A _{2},\dots , A _{ n }$? 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 $f _{i}$'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.

Détail

Keywords: elimination theory, resultant, product formula, Newton polytope