Factorizations of some generalized exponential polynomials

Abstract

In 1927, Ritt proved that a complex exponential sum can be uniquely factored as a product of irreducible and simple parts. The first part of this thesis deals with the problem of enlarging the three possible sets of elements involved in Ritt's factorization theorem, namely, coefficients, exponents and exponential function. This is done by analyzing Ritt's original proof. The Skolem-Mahler-Lech Theorem states that if an exponential polynomial has infinitely many integer zeros, then all but finitely many such zeros form a finite union of arithmetic progressions. Based on this result, Shapiro in 1959, established a factorization theorem for such exponential polynomials. Allowing the exponents in the exponential polynomial to be integer polynomials, the Skolem-Mahler-Lech theorem still holds for a certain subclass of this set. In the second part of this thesis, a factorization theorem, in the spirit of Shapiro's result, is proved for some elements of this subclass.