### Graduate Student Seminar, 5:00pm April 5, 2004; HH127

#### Speaker:

Daniel Bates
#### Title:

A small dose of polynomial culture and history.
#### Abstract:

The solution of polynomial systems has been a driving force behind
mathematics for centuries. This talk will be a survey of several of the many
techniques developed over the years for counting and locating the roots of
polynomial systems. We will begin with some elegant classical results dating
from the 17th and 19th centuries for estimating both the number and the
location of real roots of a single univariate polynomial. We will then
generalize to the more recent (1970's) Bernstein theorem which gives a bound
(the BKK bound) on the number of isolated solutions of a square polynomial
system. From there, given time, we will discuss briefly the most general mxn
case which may be handled by homotopy continuation (1980's and beyond). Along
the way we will hit a variety of numerical speedbumps, at which point we will
slow down and take a closer look at some of the many numerical woes one
encounters when working with polynomial systems.

To volunteer to give a talk, or for any other questions regarding this schedule, contact Wesley Calvert