Departmental Colloquium
Frank Sottile
Texas A&M University
Bounds for real solutions to structured polynomial systems
Abstract: 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.
The best-known upper bound is due to Khovanskii and is unrealistically large.
Lower bounds are a very recent phenomenon, having arisen with real Gromov-Witten invariants and (separately) with the
Wronski map in Schubert calculus. The results, though, are
striking: "Most" rational curves of degree d interpolating
3d-1 real points in the plane are real, and every rational
function with only real critical points is real.
In this talk, I will describe this background and then
discuss recent work giving upper and lower bounds on the
numbers of real solutions to some sparse polynomial systems.
This is joint work with Soprunova (lower bounds) and with Bihan and Bates (upper bounds).
Friday October 26, 2007 at 3:00 PM in SEO 636