Extreme points and saturated polynomials

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 (0) = 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 T²-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.