Algebraic aspects of rational tetra-inner functions

Abstract

The tetrablock E = {x ∈ C 3 : 1 − x1z − x2w + x3zw 6= 0 for |z| ≤ 1, |w| ≤ 1} has very interesting complex-geometric properties. It meets R3 in a regular tetrahedron and its distinguished boundary is homeomorphic to D × T, where D is the closed unit disc and T is the unit circle. We exploit this geometry to develop an explicit and detailed structure theory for the rational maps from the unit disc D to E, the closure of E, that maps the boundary of the disc to the distinguished boundary of E. We call such maps rational E-inner functions or rational tetra-inner functions. In this thesis, we provide a description of all rational inner functions x from D to E of degree n. Here deg(x) is the degree of x, defined in a natural way by means of fundamental groups. We show that, for any rational E-inner function x = (x1, x2, x3), deg(x) is equal to deg(x3) (in the usual sense) of the finite Blaschke product x3. The variety RE = {(x1, x2, x3) ∈ E : x1x2 = x3} plays a crucial role in the function theory of E. We prove that if x is a rational E-inner function, then either x(D) = RE or x(D) meets RE exactly deg(x) times. For a rational E-inner function x, we call the points λ ∈ D such that x(λ) ∈ RE the royal nodes of x. We describe the construction of rational E-inner functions x = (x1, x2, x3) of prescribed degree from the following interpolation data: the zeros of x1 and x2 in D and the royal nodes of x. It is easy to see that the set J of all rational E-inner functions is not convex. We prove that the subset of J of rational E-inner functions (x1, x2, x3) for a fixed inner function x3 is convex. We show that a rational E-inner function x is not an extreme point of the set J if the number of royal nodes of x on T, counted with multiplicity, is less than or equal to 1 2 deg(x).