Bounds for real solutions to equations from geometry

Frank Sottile

   Understanding the real solutions to systems of polynomial
equations  is a difficult  question with  many applications.
In  particular, a  non-trivial lower  bound is  an existence
proof  for  solutions  and  non-trivial  upper  bounds  give
complexity bounds.

   While  it  is hopeless  to  expect  anything in  general,
recent work  gives many  examples of equations  with special
combinatorial or  geometric structure possessing non-trivial
bounds.

   In this  talk, I will survey some  of these developments,
including the striking  results using tropical geometry that
"most" rational  curves of degree d  interpolating 3d-1 real
points  in the  plane are  real  and the  resolution of  the
Shapiro conjecture  which implies, for  instance, that every
rational function with only real critical points is real.  I
will  also  discuss intriguing  lower  bounds and  realistic
upper  bounds   recently  obtained  for   sparse  polynomial
systems.