An Investigation in to the Properties of Functions Defining Distinguished Varieties

Abstract

An inner toral polynomial is a polynomial in \(\mathbb{C}\) [z,w] such that its zero set is contained in \(\mathbb{D}\)2 \(\cup\) \(\mathbb{T}\)2 \(\cup\) \(\mathbb{E}\)2 where \(\mathbb{D}\) is the open unit disc, \(\mathbb{T}\) is the unit circle and \(\mathbb{E}\) is the exterior of the closed unit disc in \(\mathbb{C}\). Given such a polynomial p, it's zero set that lies inside \(\mathbb{D}\)2 , i.e V = Z (p) \(\cap\) \(\mathbb{D}\)2 is called a distinguished variety, and p is called a polynomial defining the distinguished variety V . An inner toral polynomial always gives a distinguished variety and vice versa. Finite Blaschke products generate inner toral polynomials such a way that, given a finite Blaschke product B(z) the numerator of wm - B(z) is an inner toral polynomial. In this paper, we investigate the conditions that make the sum and the composition of inner toral polynomials generated by finite Blaschke products, inner toral.