SPEAKER: Frank Sottile (Clay Mathematical Institute and MSRI) <br>

<br>

TITLE: Lower bounds for real polynomial systems from sign-imbalanced
       posets  <br>

<br>

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. This is joint work with
    Evgenia Soprunova.