Mixed discriminants

My talk will summarize recent joint work with different coauthors
about systems of n Laurent polynomials in n unknowns with fixed
support sets. For special choices of the coefficients, two or more of
their common solutions come together  and create a point of higher
multiplicity. The conditions under which this happens are encoded in
an irreducible polynomial in the coefficients called the mixed
discriminant, whose zero locus is the variety of (numerically)
ill-posed systems. Its degree is a piecewise linear function.
In the case of plane curves, we will give an explicit degree formula
with combinatorial ingredients and  an application to determine the
number of common real roots.