Lower bounds in real algebraic geometry and orientability of real toric varieties

TL;DR

This paper investigates the orientability of real toric varieties and how it provides lower bounds for real solutions to sparse polynomial systems, extending previous characterizations to broader cases.

Contribution

It offers a new characterization of when real toric varieties are orientable, strengthening prior results and enabling better bounds on real solutions of polynomial systems.

Findings

Characterization of orientability conditions for real toric varieties

Link between orientability and lower bounds for real solutions

Extension of Nakayama and Nishimura's work to broader classes

Abstract

The real solutions to a system of sparse polynomial equations may be realized as a fiber of a projection map from a toric variety. When the toric variety is orientable, the degree of this map is a lower bound for the number of real solutions to the system of equations. We strengthen previous work by characterizing when the toric variety is orientable. This is based on work of Nakayama and Nishimura, who characterized the orientability of smooth real toric varieties.