Baksa V. Properties of analytical vector-functions of bounded L-index in a two-dimensional ball

In the thesis, the object of investigation is the analytical vector-functions, both in a single two-dimensional ball in C^2 ball, and in the whole space C^p for arbitrary n. The first section of the dissertation contains an overview of the main results of the predecessors on the topic of the dissertation research. The second section establishes theorems that contain the necessary and sufficient conditions for the boundedness of the L-index of analytics in a unit two-dimensional ball in C^2 vector-functions in terms of locally regular behavior of their partial derivatives. The third section establishes theorems that contain both sufficient conditions and necessary conditions for the boundedness of the L-index of analytic in a unit two-dimensional ball in C^2 vector-functions, in terms of locally regular behavior of the maximum norm of the analytical vector-function on the be-disks. In the fourth subsection, the main content is to prove the analogue of Hayman's theorem, which gives a relatively simple apparatus for establishing the boundedness of the index of analytical solutions of differential equations. One criterion is derived from this theorem, which characterizes the boundedness of the L-index in terms of the sums of partial derivatives. Although the theorem has the character of a criterion, it “hides” an even thinner property of the power series of the analytic vector-function F in the unit ball of bounded L-index. In fact, the boundedness of such an index is equivalent to existence, the so-called main polynomial. And proving this fact is the main content of the sixth section. The seventh section is devoted to the study of the possible growth rate of analytical vector functions F in the unit ball of the bounded L-index. In particular, the upper estimate of the growth rate of all solutions is given. The latter circumstance allows us to optimistically expect effective applications of the research conducted in the work to the analytical theory of differential equations. Section 3 is devoted to the establishment of an analogue of the one-dimensional Fricke criterion of the boundedness of the index of an integer function from one variable in the class of integers vector of functions F from C^n to C^p.