Afton H. Cayford

Research Interests

Several complex variables; Entire functions; Growth rate; Non-archimedian variables

Research in Progress

In 1915 P{\accent 18 o}lya related the values of an entire function and its derivatives at a point to its growth rate in a well known theorem:
If f(z) is a transcendental entire function and f^{(n)}(0) is integral for n=0,1,... then f(z) is at least of order 1 type 1.
Since then many theorems of this type have been developed. The set of values has been enlarged, the point at which the function values are taken on has been increased to be a given set or region in the plane, the function has been replaced by a set of fu nctions, related by some condition like membership in a differential ring, and so on. Recently, theorems of this kind have been given in terms of a Non-archimedian variable and I am extending my previous work into this area. The metric properties are enti rely different but the concept of relating growth to the characteristics of the function on certain sets or to the maximal term in a power series expansion seems to be convertible.

Meromorphic functions have many of the properties of entire functions an d I have recently considered growth rate in the neighbourhood of a singularity, instead of at infinity, in this variable. A variation of previously used techniques is applicable and has been used successfully Also, after much extension in the direction of sheaf theory and the application of algebraic tools, work in several complex variables is developing in terms of classical integral representation theory incorporating these new ideas. Integral representatio ns are exactly suited to estimations of growth rate and are often expressed in terms of function values at selected points in the domain of the function. The Schnierlman Integral, recently defined, seems particularly useful.

Single variable results sugg est that a set of functions with a restricted growth rate and some restrictions on certain of their value sets will have a common characteristic, like differential dependence. This should also hold true in the several variable case although the fact that zeroes of such functions are no longer isolated requires substantive changes to the choice of subsets of the domain. Bombieri's work in describing the set of points at which a certain collection of meromorphic functions of several complex variables takes on values in a specific number field as an algebraic hypersurface has suggested methods which might be used to extend my single variable results into several variables. My recent work has been in this area.

Selected Publications