Uniqueness of Non-archimedean Entire Functions Sharing Sets of Values Counting Multiplicity

A set is called a unique range set for a certain class of functions if each inverse image of that set uniquely determines a function from the given class. We show that a finite set is a unique range set, counting multiplicity, for non-Archimedean entire functions if and only if there is no non-trivial affine transformation preserving the set. Our proof uses a theorem of Berkovich to extend, to non-Archimedean entire functions, an argument used by Boutabaa, Escassut, and Haddad to prove this result for polynomials A well-known theorem of R. Nevanlinna (cf. [Nev]) says that if f and g are two meromorphic functions such that f−1(ai) = g−1(ai) for five distinct points a1, . . . , a5 on the Riemann sphere, then either both f and g are constant, or f ≡ g. A similar result, but with only two values ai, and with meromorphic functions replaced by polynomials, can be found in [A-S], where Adams and Straus also show that the statement remains true if f and g are even allowed to be non-Archimedean entire functions. Throughout this work, the expression non-Archimedean entire function will mean a formal power series in one variable with coefficients in an algebraically closed field K of characteristic zero, complete with respect to a (possibly trivial) non-Archimedean absolute value, and such that the power series has infinite radius of convergence. This theorem of Adams and Straus fits into a principle of the first author of the present work, which states that most theorems that are true for polynomials will also be true for non-Archimedean entire functions, if stated appropriately. See [Ch] for a geometric conjecture based, in part, on this principle. The theorem of Nevanlinna quoted above says that if f−1(ai) = g−1(ai) for some values ai, then f and g must be equal. Rather than consider the values one at a time, we can weaken the hypothesis by considering the values together as a set. Namely, given a set of values S, we say that two functions f and g share S, ignoring multiplicity, if f−1(S) = g−1(S). One can also take multiplicity into account. Namely, if given a function f, we let E(f, S) = {(z, m) ∈ K × Z : f(z) = a ∈ S and f(z) = a with multiplicity m}, Received by the editors July 18, 1997. 1991 Mathematics Subject Classification. Primary 11S80, 30D35. Financial support for the first author was provided by National Science Foundation grants DMS-9505041 and DMS-9304580. The second author’s research was partially supported by a UGC grant of Hong Kong. c ©1999 American Mathematical Society