Bounds for real solutions to sparse polynomial systems

   Understanding the real solutions to systems of polynomial
equations  is a difficult  question with  many applications.
In this talk,  I will discuss recent work  with Soprunova in
which we prove the existence of lower bounds for many sparse
polynomial systems,  and give  a method for  computing these
lower bounds in some  instances.  I will also discuss recent
work with  Bertrand and Bihan giving  realistic upper bounds
for  the  number of  real  solutions  to polynomial  systems
supported on circuits.