Alain Escassut

On uniqueness of p-adic entire functions

Abstract Let K be a complete ultrametric algebraically closed field, let A (K) be the ring of entire functions in K. Unique Range Sets ( urss ) were defined in [5], and studied in [6] for complex entire or meromorphic functions. Here, we characterize the urss for polynomials, in any algebraically closed field, and we prove that in non archimedean analysis, there exist urss of n elements, for entire functions, for any n ≥ 3. When n = 3, we can characterize the sets of three elements that are urss for entire functions.

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

and 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 eld of fractions of the ring of bounded analytic functions in the disc.

Ultrametric Banach Algebras

Tree structure ultrametric absolute values Hensel lemma circular filters analytic elements holomorphic properties classic partitions holomorphic functional calculus pseudo-density definition of affinoid algebras Jacobson radical of affinoid algebras separable fields Krasner-Tate algebras universal generators in Tate algebras associated idempotents. (Part contents)

Eléments analytiques et filtres percés sur un ensemble infraconnexe

SummaryLet K be a complete ultrametric algebraically closed field, let D be an infraconnected bounded closed set of K and let H(D) be the Banach algebra of the analytical elements on D. The properties of the elements f of H(D) are learnt introducing a function v(f, μ) continuous and affine on pieces in the intervals where it is bounded. We learn the elements f ε H(D) which are not a product of a polynomial with an invertible element. We introduce the notion of monotonous filter, related with the continuous multiplicative semi-norms of H(D) and we prove these such elements are annulated by a monotonous filter and more precisely, a pierced monotonous filter.

Nevanlinna Theory in Characteristic P and Applications

Let K be a complete ultrametric algebraically closed field of characteristic p. We show that Nevanlinna’s main Theorem holds, with however some corrections. Then, many results obtained in characteristic zero have generalization. When p ≠ 0, we have to make new proofs in most of the cases. Many algebraic curves admit no parametrizations by meromorphic functions in K, or by unbounded meromorphic functions inside a disk, like in zero characteristic, provided we assume one of the function to have a non zero derivative. More generally, certain functional equations have no solution. In zero characteristic, results previously obtained are somewhat generalised, and then are extended to any characteristic. In functional equations f m + g n = 1, conclusions also are similar to those obtained in zero characteristic, provided we replace m, n by \( \tilde m = m|m{|_p} \) , n = n|n| p . We consider the Yoshida Equation in charactersitic p ≥ 0 and characterise all solutions when it has constant coefficients: this generalizes previous results in characteristic zero but with a more general form involving polynomials with a zero derivative. Proofs are given in a preprint where applications to the abc-problem are also obtained.

Uniqueness of p-adic meromorphic functions

Abstract Let K be a complete ultrametric algebraically closed field of characteristic zero, and let M ( K ) be the field of meromorphic functions in K . For all set S in K and all f ɛ M ( K ), we denote by E ( f,S ) the subset of K × : U a ɛ S ( z,g ) ɛ K × * ¦ z zero of order q of f − a . After studying, in a previous article, unique range sets for entire functions in K , we consider here a similar problem for meromorphic functions by showing, in particular, that, for every n ≥ 5, there exist sets S of n elements in K such that if f , g ɛ M ( K ) have the same poles (counting multiplicities) and satisfy E ( f, S ) = E ( g, S ), then f = g . We show how to construct such sets.

On uniqueness of

Let K be a complete ultrametric algebraically closed field of characteristic zero, and let M(K) be the field of meromorphic functions in K. For all set S in K and for all f E M(K) we denote by E(f, S) the subset of KxN*: U-E S(z, q) E KxN* I z zero of order q of f (z) a}. After studying unique range sets for entire functions in K in a previous article, here we consider a similar problem for meromorphic functions by showing, in particular, that, for every n > 5, there exist sets S of n elements in K such that, if f, g E M(K) have the same poles (counting multiplicities), and satisfy E(f, S) = E(g, S), then f = g. We show how to construct such sets. INTRODUCTION AND THEOREMS Notation. K will denote a complete ultrametric algebraically closed field of characteristic zero, and we denote by K the one dimensional projective space over K: K = KU {oo}. Given a field L, L* will denote L \ {O}. We denote by A(K) the ring of entire functions in K and by M(K) the field of meromorphic functions in all K. For a subset S of K and f E M(K) we denote by E(f, S) the set in KxN*: U {(z,q) E KxN*I z zero of order q of f(z)-a}. aES Besides, given a subset of K containing {oo}, we denote by E(f, S) the subset of KXN* : E(f, S n K) U {(z, q) Iz pole of order q of f }. Definition. Let Y be a nonempty subset of M(K). A subset S of K is called a unique range set (a URS in short) for F if for any f, g E .F such that E(f, S) = E(g, S), one has f = g. In the same way, a couple of sets S, T in K such that SnT = 0 will be called a biURS for Y if for any f, g E Y such that E(f, S) = E(g, S) and E(f, T) = E(g, T), one has f = g. Remark 1. If a set S is a URS for A(K) (resp. M(K)), then for every nonconstant affine (resp. partial rational linear) function h, h(S) also is a URS for A(K) (resp. for M(K)). In the same way, if a couple of sets (S, T) is a bi-URS for A(K) (resp. Received by the editors October 22, 1996 and, in revised form, December 10, 1996 and January 31, 1997. 1991 Mathematics Subject Classification. Primary 11Q25. ?)1998 American Mathematical Society

p

Let be a complete ultrametric algebraically closed field of characteristic 0. According to the p-adic Hayman conjecture, given a transcendental meromorphic function f in , for each , takes every value infinitely many times. It was proven by the second author for . Here, we prove it for by using properties of meromorphic functions having finitely many multiple poles.

-adic meromorphic functions

Let K be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value | · |. Given a meromorphic function f in K (resp. inside an ‘open’ disk D) we check that the field of small meromorphic functions in K (resp. inside D) is algebraically closed in the whole field of meromorphic functions in K (resp. inside D). If two analytic functions h, l in K, other than affine functions, satisfy h′l − hl′ = c ∈ K, then c = 0. The space of the entire functions solutions of the equation y″ = φy, with φ a meromorphic function in K or an unbounded meromorphic function in D, is at most of dimension 1. If a meromorphic function in K has no multiple pole, then f′ has no exceptional value. If f is meromorphic with finitely many zeroes then for every c ≠ 0, f′ − c has an infinity of zeroes. If is not a constant or an affine function and if f has no simple pole with a residue equal to 1, then f′ + f 2 admits at least one zero. When K has residue characteristic zero, we extend some results for entire functions to analytic functions in D.

The p-adic Hayman conjecture when n = 2

Q 1. Composantes infraconnexes. Couronnes vides Pour pouvoir conclure de fagon definitive sur les relations existant entre la forme d’un fermd born4 D et le fait que H(D) soit une algebre de Banaoh noetherienne, nous sommes amen& a etudier H(D) lorsque D n’est pas infraconnexe puisque le probkme vient d’etre resolu lorsque D est infraconnexe. Pour cela nous devons Studier d’abord quelques propriMs relatives precis6ment 8, l’infraconnexit6. Rappelons qu’une condition necessaire et suffisante pour qu’un ferme borne D soit infraconnexe est que, quel que soit a ED et pour tout nombre reel r compris entre 0 et le diametre R de D, il existe une suite de points distincts xn de D tels que lirn+,lxn--al =T. Cette proprieti Bquivalente va nous permettre de definir la notion de composantes infraconnexes de D.