A history of psd and sos polynomials (before the work of the speaker and his host)

Algebra Seminar
Monday, November 4, 2013 - 3:00pm for 1 hour (actually 50 minutes)
Skiles 005
Bruce Reznick – University of Illinois, Urbana-Champaign
Greg Blekherman
A real polynomial is called psd if it only takes non-negative values. It is called sos if it is a sum of squares of polynomials. Every sos polynomial is psd, and every psd polynomial with either a small number of variables or a small degree is sos. In 1888, D. Hilbert proved that there exist psd polynomials which are not sos, but his construction did not give any specific examples. His 17th problem was to show that every psd polynomial is a sum of squares of rational functions. This was resolved by E. Artin, but without an algorithm. It wasn't until the late 1960s that T. Motzkin and (independently) R.Robinson gave examples, both much simpler than Hilbert's. Several interesting foundational papers in the 70s were written by M. D. Choi and T. Y. Lam. The talk is intended to be accessible to first year graduate students and non-algebraists.