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An ultrametric Nevanlinna’s second main theorem for small functions of a special type

Henna Jurvanen (2010)

Annales mathématiques Blaise Pascal

In ultrametric Nevanlinna theory, the Nevanlinna’s second main theorem for small functions has only been proved in the case of at most three small functions. In this paper, we prove a second main theorem for q small functions of a special type when the residue characteristic of the field is zero.

Analyse p -adique

Yvette Amice (1959/1960)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

Analytic potential theory over the p -adics

Shai Haran (1993)

Annales de l'institut Fourier

Over a non-archimedean local field the absolute value, raised to any positive power α > 0 , is a negative definite function and generates (the analogue of) the symmetric stable process. For α ( 0 , 1 ) , this process is transient with potential operator given by M. Riesz’ kernel. We develop this potential theory purely analytically and in an explicit manner, obtaining special features afforded by the non-archimedean setting ; e.g. Harnack’s inequality becomes an equality.

Applications of the p -adic Nevanlinna theory to functional equations

Abdelbaki Boutabaa, Alain Escassut (2000)

Annales de l'institut Fourier

Let K be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We apply the p -adic Nevanlinna theory to functional equations of the form g = R f , where R K ( x ) , f , g are meromorphic functions in K , or in an “open disk”, g satisfying conditions on the order of its zeros and poles. In various cases we show that f and g must be constant when they are meromorphic in all K , or they must be quotients of bounded functions when they are meromorphic in an “open disk”. In particular,...

Casoratien et équations aux différences p -adiques

Jean-Paul Bézivin (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

Dans cet article, nous démontrons une inégalité liant la croissance d’un casoratien généralisé de m séries entières p -adique à la croissance du casoratien ordinaire de ces m séries entières. Il en résulte que si le casoratien de m fonctions entières p -adiques est un polynôme non nul, alors toutes ces fonctions sont des polynômes. Comme application, nous montrons que si une équation aux différences linéaire d’ordre t à coefficients dans p [ x ] a t solutions méromorphes dans tout p , linéairement indépendantes...

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