On unique range sets for polynomials or rational functions
by
Kazuya Tohge
Kanazawa University, Faculty of Engineering
Coauthors: Gary G. Gundersen (University of New Orleans)

Let F be a family of non-constant meromorphic functions defined in the complex plane C. A set S subset C is called a unique range set for this family F , if any two functions in F having the same preimages of the set S with multiplicities are identically equal on C. Many examples of (finite) unique range sets have been obtained for the families M and E of all non-constant meromorphic functions on C and of all non-constant entire functions, respectively, and we denote such a unique range set as a URSM or a URSE, respectively.

In this talk, we will consider the cases when F=R (all non-constant rational functions) and F=P (all non-constant polynomials), and we denote a corresponding unique range set as a URSR and a URSP, respectively. We will show that there is a clear difference between these families R, P and M, E. For example, C. C. Yang and X. H. Hua verified that a URSM must have at least 6 elements and a URSE must have at least 5 elements. We show that when F=P, there cannot exist any URSP with 2 elements, but almost all sets with 3 elements can be URSP's. For the case when F=R, we see that a URSR must have at least 5 elements. Some characterization and related results on URSE and URSR will be discussed also.

This is a preliminary report of joint work with Gary. G. Gundersen, University of New Orleans.

Date received: May 30, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahs-35.