# Extreme points and saturated polynomials

## TL;DR

This paper investigates the extreme points of a set of analytic functions on the bidisk with positive real part, focusing on those with rational inner Cayley transforms, and introduces saturated polynomials related to these extreme points.

## Contribution

The paper constructs families of extreme points using saturated polynomials and conjectures these are all such extreme points, advancing the understanding of function extremality in complex analysis.

## Key findings

- Constructed families of extreme points from saturated polynomials.
- Conjecture that these are all extreme points with rational inner Cayley transforms.
- Provided a characterization of extreme points in the specified function set.

## 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(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.