SR 1011

My talk will discuss the problem of solving systems of homogeneous polynomials over a field with some recent applications to quadratic form theory. I will begin with classical results due to Lang and quickly move to more recent results found by Greenberg, Pfister, Amer, Brumer, Heath-Brown, and myself. The fields include finite fields, p-adic fields, the complex numbers, function fields over these fields, and power series fields. The main question is how many variables are needed to guarantee the existence of nontrivial solutions of systems of homogeneous polynomials defined over these fields. The case of systems of quadratic forms leads to new results about u-invariants of fields. This talk will be accessible to both graduate students and faculty members.

Prof. Dr. David Leep

University of Kentucky, Lexington, USA