Analysis Seminar: "Bounded rational functions in new settings"
Abstract: The admissible numerator problem is the following: given a polynomial p with no zeros on some domain D, describe all polynomials q such that q/p is bounded on D. Joint work with Bickel, Pascoe, Sola along with a paper by Kollar solved the admissible numerator problem for the bidisk by converting it to a local problem in the context of the bi-upper half plane. A key tool was understanding Puiseux expansions for stable polynomials. Recently we have extended some elements of this theory to the unit ball in C^2, where the Puiseux series behave very differently, as well as to special cases in the tri-disk where Puiseux series are unavailable. In particular, the case of p having a smooth point of its zero set intersecting the 3-torus becomes tractable.
Host: Walton Green