Moscow Mathematical Journal

Volume 17, Issue 4, October–December 2017 pp. 717–740.

The Resultant of Developed Systems of Laurent Polynomials

Authors: A. G. Khovanskii (1) and Leonid Monin (2)
Author institution:(1) Department of Mathematics, University of Toronto, Toronto, Canada; Moscow Independent University, Moscow, Russia
(2) Department of Mathematics, University of Toronto, Toronto, Canada

Summary:

Let R_Δ(f₁, ..., f_n+1) be the Δ-resultant (defined in the paper) of (n+1)-tuple of Laurent polynomials. We provide an algorithm for computing R_Δ assuming that an n-tuple (f₂, ..., f_n+1) is developed. We provide a relation between the product of f₁ over roots of f₂ = ··· = f_n+1 = 0 in (C*)ⁿ and the product of f₂ over roots of f₁ = f₃ = ··· = f_n+1 = 0 in (C*)ⁿ assuming that the n-tuple (f₁f₂, f₃, ..., f_n+1) is developed. If all n-tuples contained in (f₁, ..., f_n+1) are developed we provide a signed version of Poisson formula for R_Δ. In our proofs we use topological arguments and topological version of the Parshin reciprocity laws.

2010 Math. Subj. Class. 14M25.

Keywords: Newton polyhedron, Laurent polynomial, developed system, resultant, Poisson formula, Parshin reciprocity laws.

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