Leonora Krajina: Higher-dimensional singularities of rational inner functions

Abstract: In this thesis, we study singularities of rational inner functions (RIFs) ϕ in three and more variables. Even though substantial research has been done in the two-variable case, much is left unknown in higher dimensions. We focus on curve and higherdimensional singularities of ϕ, where quite different behaviour is observed compared to isolated singularities. Among other things, we show that the integrability conditions of partial derivatives of ϕ under certain assumptions stay constant along irreducible components of the zero set of the function ρϕ, and that the presence of vertical line singularities does not affect global integrability, therefore refuting Question 2 in [9]. Those results are then generalized to n variables as well. We take a look at slice matrices, which we use to obtain integrability conditions of compositions ϕN. Finally, we do a quick observation of level sets and present an overview of interesting original examples, including an RIF with two vertical line singularities admitting a horizontal level surface.