The Resultant of Developed Systems of Laurent Polynomials

TL;DR

This paper introduces an algorithm for computing the $ riangle$-resultant of Laurent polynomial systems when certain tuples are developed, and establishes relations between products over roots, extending Poisson formulas with topological methods.

Contribution

It develops an algorithm for the $ riangle$-resultant of Laurent polynomials assuming tuples are developed, and relates products over roots, extending Poisson formulas with topological tools.

Findings

Provided an algorithm for computing the $ riangle$-resultant under developed conditions.

Established relations between products over roots of polynomial systems.

Extended Poisson formula for the $ riangle$-resultant using topological methods.

Abstract

Let $R_\Delta (f_1,\ldots,f_{n+1})$ be the {\it $\Delta$-resultant} (see below) of $(n+1)$-tuple of Laurent polynomials. We provide an algorithm for computing $R_\Delta$ assuming that an $n$-tuple $(f_2,\dots,f_{n+1})$ is {\it developed} (see sec.6). We provide a relation between the product of $f_1$ over roots of $f_2=\dots=f_{n+1}=0$ in $(\mathbb C^*)^n$ and the product of $f_2$ over roots of $f_1=f_3=\dots=f_{n+1}=0$ in $(\mathbb C^*)^n$ assuming that the $n$-tuple $(f_1f_2,f_3,\ldots,f_{n+1})$ is developed. If all $n$-tuples contained in $(f_1,\dots,f_{n+1})$ are developed we provide a signed version of Poisson formula for $R_\Delta$. In our proofs we use a topological arguments and topological version of the Parshin reciprocity laws.