Extreme points and saturated polynomials

Abstract

We consider the problem of characterizing the extreme points of the set of analytic functions $f$ on the bidisk with positive real part and $f\left(0\right)=1$. If one restricts to those $f$ whose Cayley transform is a rational inner function, one gets a more tractable problem. We construct families of such $f$ that are extreme points and conjecture that these are all such extreme points. These extreme points are constructed from polynomials dubbed ${\mathbb{T}}^2$-saturated, which roughly speaking means they have no zeros in the bidisk and as many zeros as possible on the boundary without having infinitely many zeros.