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


Daniel Bates


A small dose of polynomial culture and history.


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