Lower bounds for real polynomial systems from sign-imbalanced posets

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. This is joint work with Evgenia Soprunova.