Analysis Seminar: Boundedness on the edge: Singularities of rational functions in dimensions greater than 2

Abstract:  Given a rational function that is analytic on some domain, suppose it has a singularity on the boundary of the domain.  How can we tell if the function is bounded in the domain near the singularity?  A recent result of Bickel, myself, Pascoe, and Sola along with a result of Kollár solved this in the case when the domain is the bidisk.  In this talk we move to higher dimensional polydisks.  Already for the simplest case where the denominator’s zero set is a smooth variety going through the boundary point the problem is nontrivial.  We construct interesting examples to show how bad it can get and give a solution to the problem for when the singularity is isolated on the d-torus.