Rational tetra-inner functions and the special variety of the tetrablock

Abstract

The setE‾={x∈C3:1−x1z−x2w+x3zw≠0 whenever |z|<1,|w|<1} is called the tetrablock and has intriguing complex-geometric properties. It is polynomially convex, nonconvex and starlike about 0. It has a group of automorphisms parametrised by AutD×AutD×Z2 and its distinguished boundary bE‾ is homeomorphic to the solid torus D‾×T. It has a special subvarietyRE‾={(x1,x2,x3)∈E‾:x1x2=x3}, called the royal variety of E‾. RE‾ is a complex geodesic of E and it is invariant under all automorphisms of E. We make use of these geometric properties of E‾ to develop an explicit structure theory for the rational maps from the unit disc D to E‾ that map the unit circle T to the distinguished boundary bE‾ of E‾. Such maps are called rational E‾-inner functions. 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 of prescribed degree from the zeros of x1 and x2 and the royal nodes of x. The proof of this theorem is constructive: it gives an algorithm for the construction of a 3-parameter family of such functions x subject to the computation of Fejér-Riesz factorizations of certain non-negative functions on the circle. We show that, for each nonconstant rational E‾-inner function x, either x(D‾)⊆RE‾∩E‾ or x(D‾) meets RE‾ exactly deg⁡(x) times. We study convex subsets of the set J of all rational E‾-inner functions and extreme points of J. We show that whether a rational E‾-inner function x is an extreme point of J depends on how many royal nodes of x lie on T.