What question did this study set out to answer?

This research investigates the algebraic nature of derivatives of meromorphic functions within a certain disk.

April 3, 2026Mathematical Notes0 citations

On Common Algebraic Points of a Pair of Different Derivatives of Finite-Order Meromorphic Functions

Key Points

This research investigates the algebraic nature of derivatives of meromorphic functions within a certain disk.
Defined a set of points where derivatives are algebraic numbers with specific degree and height constraints.
Investigated pairs of derivatives for conditions excluding rational and elliptic forms.
Used mathematical bounding techniques to establish limits on the number of points.
Established an upper bound for the number of algebraic points for almost all combinations of parameters (n, H, R).
Concluded that the identified points have bounded degree and height with respect to meromorphic functions.

Abstract

Given nonnegative integers t<s, pairs (f^ (t) (z), f^ (s) (z) ) of derivatives of finite-order meromorphic functions f (z) for which f^ (t) (z) is neither a rational function, a rational function of an exponential e^ z, nor an elliptic function are considered. For positive integers n and H and a positive number R, let B (n, H, R) be the set of points z in the disk |z| R for which f^ (t) (z) and f^ (s) (z) are algebraic numbers of degree at most n and height at most H (and, moreover, |f^ (t) (z) | and |f^ (s) (z) | are not very large). An upper bound for the number of points in B (n, H, R) is obtained for almost all, in a certain sense, (n, H, R).

AI से पूछें

Bookmark

View Full Paper