Title:

Algebraic aspects of rational tetrainner functions

The tetrablock E = {x ∈ C 3 : 1 − x1z − x2w + x3zw 6= 0 for z ≤ 1, w ≤ 1} has very interesting complexgeometric 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 Einner functions or rational tetrainner 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 Einner 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 Einner function, then either x(D) = RE or x(D) meets RE exactly deg(x) times. For a rational Einner function x, we call the points λ ∈ D such that x(λ) ∈ RE the royal nodes of x. We describe the construction of rational Einner 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 Einner functions is not convex. We prove that the subset of J of rational Einner functions (x1, x2, x3) for a fixed inner function x3 is convex. We show that a rational Einner 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).
