Analysis Seminar: Local square integrability of rational functions in two variables

Abstract: Given a polynomial p(x,y) with no zeros in the bidisk (|x|,|y|<1) we are interested in boundary singularities of rational functions Q(x,y)/p(x,y). There are many ways to study the nature of a boundary singularity but in this talk we will discuss square-integrability on the two torus (|x|=|y|=1 ) of the rational function. This problem converts to a local square integrability question on R^2 near (0,0) after applying conformal maps. After doing this we are able to give a complete characterization of the locally square integrable rational functions in two variables (with denominator non-vanishing on the product upper half plane). Time permitting we will discuss an application to sums of squares decompositions for stable polynomials.