Summary |
In this thesis, we shall prove some results which, in turn, will allow us to solve some factorization problems in a systematic way. Also, we shall utilize new methods from theory of complex analytic sets and local holomorphic dynamlics to solve some factorization problems. In Chapter 3, by using an extended version of Steinmetz's theorem, we prove that certain class of meromorphic functions is pseudo-prime. Hence, we can prove that under certain conditions, R(z)H(z) is pseudo-prime, where R(z) is a non-constant rational function and H(z) is a finite order periodic function. In Chapter 4, we try to find out all possible factorizations of p(z)H(z) when H is an exponential type periodic function and p is a non-constant polynomial. This confirms a conjecture of G.D. Song and C.C. Yang. In Chapter 5, we shall use results from theory of complex analytic sets to prove certain criteria on the existence of a non-linear entire common right factor of two entire functions. Applying these criteria, we can then prove that if f is an entire function which is pseudo-prime and not of the form H(Q(z)), where H is a periodic entire function and Q is a polynomial, then R(f(z)) is also pseudo-prime for any non-constant rational function R. This result essentially solves a problem of G.D. Song and is a fundamental property of pseudo-prime function. We also give other applications of these criteria to unique factorization problems. In Chapter 6, we consider the unique factorization problems of f o p and p o f where f is a prime transcendental entire function and p is prime polynomal. We shall use methods from local holomorphic dynamics to solve these problems. |