Lattice Polygons and Real Roots
It is known from theorems of Bernstein, Kushnirenko and Khovanskii
from the 1970s that the number of complex solutions of a system of
multivariate polynomial equations can be expressed in terms of
subdivisions of the Newton polytopes of the polynomials. For very
special systems of polynomials Soprunova and Sottile (2006) found
an analogue for the number of real solutions. In joint work with
Ziegler we could give a simple combinatorial formula for the
signature of foldable triangulation of a lattice polygon. Via the
Soprunova-Sottile result this translates into lower bounds for the
number of real roots of certain bivariate polynomial systems.