Algebraic Properties of Entire Functions with Coefficients in Particular Valued Fields

Abstract

The study of entire functions is of central importance in complex function theory. We consider the ring of entire functions either on subfields of C or on some subfields of Cp . By using a technique based on admissible filters we study the ideal structure of the ring of entire functions. Then we prove the B ́zout property for the ring of entire e functions over Cp independent of Mittag-Leffler theorem. An important problem in complex function theory is to find an entire function from its values on a given sequence. By means of so-called Newton entire functions we solve a series of interpolation problems. Then we obtain a general result which implies the results of P ́lya and Gel’fond on the entire functions which are polynomials. We o prove a similar result for the entire functions f such that f (D) ⊂ D, where D is a particular bounded set. As an application we replace the use of power series for the initial value problems for ODE’s with Newton series for boundary value problems.