Bounds on the number of real solutions to polynomial equations

TL;DR

This paper establishes an upper bound on the number of non-zero real solutions to certain polynomial systems using Gale duality, improving understanding of solution counts in algebraic geometry.

Contribution

It introduces a new bound for real solutions of polynomial systems with specific monomial and subgroup conditions, extending previous fewnomial bounds.

Findings

Bound exceeds positive solution bound only by a constant factor

Bound is asymptotically sharp for fixed k and large n

Uses Gale duality to adapt existing proof techniques

Abstract

We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of Z^n of odd index. This bound exceeds the bound for positive solutions only by the constant factor (e^4+3)/(e^2+3) and it is asymptotically sharp for k fixed and n large.