An Investigation in to the Properties of Functions Defining Distinguished Varieties

An inner toral polynomial is a polynomial inC[z,w] such that its zero set is contained inD2∪T2∪E2, whereDis the open unit disc,Tis the unit circle andEis the exterior of the closed unit disc inC. Given such apolynomialp, it’s zero set that lies insideD2, i.eV=Z(p)∩D2is called a distinguished variety, andpis calleda polynomial defining the distinguished varietyV. An inner toral polynomial always gives a distinguishedvariety and vice versa. Finite Blaschke products generate inner toral polynomials such a way that, given afinite Blaschke productB(z), the numerator ofwm−B(z) is an inner toral polynomial. In this paper, weinvestigate the conditions that make the sum and the composition of inner toral polynomials generated byfinite Blaschke products, inner toral.