# 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}(*f*_{1}, ..., *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 (*f*_{2}, ..., *f*_{n+1}) is developed.
We provide a relation between the product of *f*_{1} over roots of *f*_{2} = ··· =
*f*_{n+1} = 0 in (**C***)^{n} and the product of *f*_{2} over roots of *f*_{1} = *f*_{3} = ··· =
*f*_{n+1} = 0 in (**C***)^{n} assuming that the *n*-tuple (*f*_{1}*f*_{2}, *f*_{3}, ..., *f*_{n+1}) is
developed. If all *n*-tuples contained in (*f*_{1}, ..., *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.

Contents Full-Text PDF