Unicity theorems for meromorphic or entire functions II

Abstract

In 1976, Gross posed the question “can one find two (or possibly even one) finite sets S j ( j = 1, 2) such that any two entire functions f and g satisfying E f ( S j ) = E g ( S j ) for j = 1,2 must be identical?”, where E f ( S j ) stands for the inverse image of S j under f . In this paper, we show that there exists a finite set S with 11 elements such that for any two non-constant meromorphic functions f and g the conditions E f ( S ) = E g ( S ) and E f ({∞}) = E g ({∞}) imply f ≡ g . As a special case this also answers the question posed by Gross.