Optimal Descartes' Rule of Signs for Circuits

TL;DR

This paper introduces an optimal version of Descartes' rule of signs that precisely bounds the positive real roots of sparse polynomial systems with n+2 monomials, based on sign variation analysis.

Contribution

It provides the first sharp upper bound for positive roots of such systems, improving upon previous bounds by leveraging sign variation of exponent and coefficient sequences.

Findings

Established an optimal bound for positive roots in sparse polynomial systems.

Connected the bound to sign variation of exponent and coefficient sequences.

Demonstrated the bound's sharpness and applicability.

Abstract

We present an optimal version of Descartes' rule of signs to bound the number of positive real roots of a sparse system of polynomial equations in n variables with n+2 monomials. This sharp upper bound is given in terms of the sign variation of a sequence associated to the exponents and the coefficients of the system.