The sharp bound for the number of real solutions to polynomial equation systems

TL;DR

This paper establishes the exact maximum number of positive solutions for real polynomial systems, using geometric and homotopic methods, resolving a long-standing open problem in real algebraic geometry.

Contribution

It provides the first sharp bounds for the number of positive solutions in real polynomial systems, extending known results to both unmixed and mixed cases with support-based formulas.

Findings

Sharp bound for positive solutions in unmixed systems derived from simplex triangulations.

Maximal positive solutions in mixed systems expressed as a symmetric multilinear function.

Proof utilizes homotopic arguments and inductive triangulation of polynomial supports.

Abstract

This paper solves the open problem on the sharp bound for the number of isolated solutions in $\mathbf{R}_*^n$ to the real system of $n$ polynomial equations in $n$ variables, i.e., the real $n$ by $n$ fewnomial system. For an unmixed system of $n$ polynomial equations in $n$ variables, this paper shows that the number of its positive solutions in $\mathbf{R}_*^n$ is sharply bounded by that of the simplex configurations in the triangulation of its support generically. The proof is based on a homotopic argument and an inductive triangulation of the support of the system via a hierarchy of pyramid configurations of different orders. For the mixed system of $n$ polynomial equations in $n$ variables, this paper shows that the maximal number of positive solutions in $\mathbf{R}_*^n$ to the systems with the same support is a symmetric multilinear function of the support generically and hence can be computed via the polarization identity.