Lower bounds for real polynomial systems from sign-imbalanced posets.Algebra & Discrete Mathematics
|Frank Sottile, Clay Mathematical Institute, MSRI, and Texas A&M University
|Fri, May 28 2004, 4:10PM
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. This is joint work with Evgenia Soprunova