Title: Lower bounds in enumerative real algebraic geometry from sign-imbalanced posets.
Session Name: AMS Special Session on Geometry and Combinatorics

Author: Evgenia Soprunova
Author: Frank Sottile

Abstract: We present families of sparse polynomial systems having a lower bound on their number of real solutions. Each family is unmixed with Newton polytope the order polytope of a finite poset P that is ranked (mod 2) and whose maximal chains have equal length (mod 2). The lower bound is the sign-imbalance of the poset---this is the difference between the number of even and of odd linear extensions of the poset P. The sign-imbalance is interpreted as the topological degree of a certain folding map of an associated simplicial complex.
    Our tools are combinatorics of toric varieties, toric degenerations, and some topology. Using sagbi degenerations, we recover results of Eremenko and Gabrielov on the degree of the Wronski map.