Bounds for real solutions to polynomial systems

  Frank Sottile

    Understanding, finding, or even deciding the existence of real solutions to 
a system of equations is a very difficult problem with many applications. While 
it is hopeless to expect much in general, we know a surprising amount about these 
questions for systems which possess additional structure.

    We will focus on equations from toric varieties and homogeneous spaces, particularly
Grassmannians. Not only is much known in these cases, but they encompass some of the
most common applications. The results we discuss may be grouped into two themes: 
  (1) Upper bounds on the number of real solutions
  (2) Lower bounds on the number of real solutions
Upper bounds as in (1) bound the complexity of the set of real solutions.
Lower bounds as in (2) give an existence proof for real solutions.