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Les semi-normes multiplicatives sur les algebres d'elements analytiques au sens de krasner

$T$-filtres, ensembles analytiques et transformation de Fourier $P$-adique

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

methods

/pdf/on-uniqueness-of-p-adic-meromorphic-functions-sqni3zg1se.pdf

On uniqueness of p-adic meromorphic functions

Let 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 e 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.

https://link.springer.com/content/pdf/10.1007/BF02418012.pdf

Eléments analytiques et filtres percés sur un ensemble infraconnexe

The ultrametric spectral theory

Let K be a complete ultrametric algebraically closed field and let A be the K -Banach algebra of bounded analytic functions in the disk D : | x | 1 . Let Mult ( A , ‖ ⋅ ‖ ) be the set of continuous multiplicative semi-norms of A , let Mult m ( A , ‖ ⋅ ‖ ) be the subset of the ϕ ∈ Mult ( A , ‖ ⋅ ‖ ) whose kernel is a maximal ideal and let Mult a ( A , ‖ ⋅ ‖ ) be the subset of the ϕ ∈ Mult m ( A , ‖ ⋅ ‖ ) whose kernel is of the form ( x − a ) A , a ∈ D (if ϕ ∈ Mult m ( A , ‖ ⋅ ‖ ) ∖ Mult a ( A , ‖ ⋅ ‖ ) , the kernel of ϕ is then of infinite codimension). The main problem we examine is whether Mult a ( A , ‖ ⋅ ‖ ) is dense inside Mult m ( A , ‖ ⋅ ‖ ) with respect to the topology of simple convergence. This a first step to the conjecture of density of Mult a ( A , ‖ ⋅ ‖ ) in the whole set Mult ( A , ‖ ⋅ ‖ ) : this is the corresponding problem to the well-known complex corona problem. We notice that if ϕ ∈ Mult m ( A , ‖ ⋅ ‖ ) is defined by an ultrafilter on D , ϕ lies in the closure of Mult a ( A , ‖ ⋅ ‖ ) . Particularly, we shaw that this is case when a maximal ideal is the kernel of a unique ϕ ∈ Mult m ( A , ‖ ⋅ ‖ ) . Thus, if every maximal ideal is the kernel of a unique ϕ ∈ Mult m ( A , ‖ ⋅ ‖ ) , Mult a ( A , ‖ ⋅ ‖ ) is dense in Mult m ( A , ‖ ⋅ ‖ ) . And particularly, this is the case when K is strongly valued. In the general context, we find a subset of Mult m ( A , ‖ ⋅ ‖ ) ∖ Mult a ( A , ‖ ⋅ ‖ ) which is included in the closure of Mult a ( A , ‖ ⋅ ‖ ) . More generally, we show that if ψ ∈ Mult ( A , ‖ ⋅ ‖ ) does not define the Gauss norm on polynomials ( ‖ ⋅ ‖ ) , then it is characterized by a circular filter, like on rational functions and analytic elements. As a consequence, if ψ does not lie in the closure of Mult a ( A , ‖ ⋅ ‖ ) , then its restriction to polynomials is the Gauss norm.

background or methods

About the ultrametric corona problem

© Publications mathématiques de l’I.H.É.S., 1962, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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Les semi-normes multiplicatives sur les algebres d'elements analytiques au sens de krasner

Citations

$T$-filtres, ensembles analytiques et transformation de Fourier $P$-adique

On uniqueness of p-adic meromorphic functions

Eléments analytiques et filtres percés sur un ensemble infraconnexe

The ultrametric spectral theory

About the ultrametric corona problem

References

Les zéros d'une fonction analytique d'une variable sur un corps valué complet

Algebres d'elements analytiques en analyse non archimedienne

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