  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 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.
