The University of Michigan Combinatorics Seminar


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 signimbalance of the posetthis is the difference between the number of even and of odd linear extensions of the poset P. The signimbalance 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. 