Bounds for the number of real solutions to systems of equations.

  Frank Sottile

  Computing, counting, or even deciding on the existence
of real solutions to a system of polynomial equations
is a very challenging problem that is important in many
applications of mathematics.   There is an emerging
landscape of structure in the possible numbers of
real solutions to systems of polynomial equations.
These include fewnomial upper bounds, gaps or congruences,
and lower bounds.  My talk will survey what is known
about these bounds, focussing on lower bounds---which
are existence proof of solutions---and open problems,
including some concrete challenges.