Finite Blaschke products versus polynomials

Abstract

The objective of the thesis is to compare polynomials and finite Blaschke products, and demonstrate that they share many similar properties and hence we can

establish a dictionary between these two kinds of finite maps for the first time.

The results for polynomials were reviewed first. In particular, a special kind of

polynomials was discussed, namely, Chebyshev polynomials, which can be defined

by the trigonometric cosine function cos ?. Also, a complete classification for two

polynomials sharing a set was given.

In this thesis, some analogous results for finite Blaschke products were proved.

Firstly, Chebyshev-Blaschke products were introduced. They can be defined by re-

placing the trigonometric cosine function cos z by the Jacobi cosine function cd(u; ? ).

They were shown to have several similar properties of Chebyshev polynomials, for

example, both of them share the same monodromy, satisfy some differential equations and solve some minimization problems. In addition, some analogous results

about two finite Blaschke products sharing a set were proved, based on Dinh's and

Pakovich's ideas.

Moreover, the density of prime polynomials was investigated in two different

ways: (i) expressing the polynomials of degree n in terms of the zeros and the leading

coefficient; (ii) expressing the polynomials of degree n in terms of the coefficients.

Also, the quantitative version of the density of composite polynomials was developed

and a density estimate on the set of composite polynomials was given. Furthermore,

some analogous results on the the density of prime Blaschke products were proved.