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

- Series
- Algebra Seminar
- Time
- Monday, November 4, 2013 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Bruce Reznick – University of Illinois, Urbana-Champaign

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.