URS AND URSIMS FOR P-ADIC MEROMORPHIC FUNCTIONS INSIDE A DISC

Abstract

Let K be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We show that the p-adic main Nevanlinna Theorem holds for meromorphic functions inside an 'open' disc in K. Let Pn,c be the Frank-Reinders's polynomialand let Sn,c be the set of its n distinct zeros. For every n 7, we show that Sn,c is an n-points unique range set (counting multiplicities) for unbounded analytic functions inside an 'open disc', and for every n 10, we show that Sn,c is an n-points unique range set ignoring multiplicities for the same set of functions. Similar results are obtained for meromorphic functions whose characteristic function is unbounded: we obtain unique range sets ignoring multiplicities of 17 points. A better result is obtained for an analytic or a meromorphic function f when its derivative is 'small' comparatively to f . In particular, for every n 5 we show that Sn,c is an n-points unique range set ignoring multiplicities for unbounded analytic functions with small derivative. Actually, in each case, results also apply to pairs of analytic functions when just one of them is supposed unbounded. The method we use is based upon the p-adic Nevanlinna Theory, and Frank-Reinders's and Fujimoto's methods used for meromorphic functions in C. Among other results, we show that the set of functions having a bounded characteristic function is just the field of fractions of the ring of bounded analytic functions in the disc.

Applications of the $p$-adic Nevanlinna theory to functional equations

“…Moreover, in order to obtain the most powerful form of the theorem, one has to keep 7V(r, /) inside the inequality, rather than replacing it prematurely by T(r, /). If we now consider meromorphic functions inside / € M{d(0,R~)), We saw in [6] that the same inequality holds and we obtain this more general theorem: THEOREM N. -Let ai,...,0g G K, and let S = {ai,...,aj. 0, R-)), the inequality is not trivial but the term -log r is no longer efficient when applying the theorem because r is now bounded.…”

URSCM OR BI-URSCM FOR p-ADIC ANALYTIC OR MEROMORPHIC FUNCTIONS INSIDE A DISK

“…And since T(r, f ) is unbounded when r tends to R, we see that 2m − n ≤ 2. Now, in the hypotheses of Theorem 2, by (5) and (6) we obtain 2m − n ≤ 3.…”

“…The concept of unique range sets counting multiplicities for a family of meromorphic functions was first introduced by F. Gross and C. C. Yang in the eighties [12]. Many papers were published on this topic and on closely related topics involving uniqueness, on complex and p-adic meromorphic functions [1], [3], [4], [5], [6], [7], [8], [10], [11], [13], [14], [16], [17].…”

Functional equations in a p-adic context

“…All over the 15 last years, many problems of uniqueness were examined, first in complex analysis and also in p-adic analysis: unique range sets [2][3][4]9,11,18,19], polynomials and functions of uniqueness [6,7,10,11,14,23], with regard to several classes of functions (entire functions, meromorphic functions in C or in a p-adic complete algebraically closed field). In the non-archimedean context, the p-adic Nevanlinna theory works not only in the whole field but also inside an open disc (concerning then unbounded meromorphic functions) [1,6,8].…”

“…The following theorem is the second main theorem in p-adic analysis and is given in [1] for M(K) and in [3] for M(d(0, R − )). Theorem A.…”