Seminar in Real & Complex Geometry

Thursday, 17.01.2013, 16:15-17:30, Schreiber building, room 210

Bernd Sturmfels, University of California, Berkeley

Non-negative polynomials versus sums of squares


We discuss the geometry underlying the difference between non-negative polynomials and sums of squares. The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are Noether-Lefschetz loci of K3 surfaces. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. Other algebraic geometry classics to appear naturally are Severi varieties, low rank Hankel matrices, and quartic symmetroids. This lecture is based on work with Blekherman, Hauenstein, Ottem and Ranestad.