By Alain Escassut

ISBN-10: 9810222343

ISBN-13: 9789810222345

The behaviour of the analytic parts on an infraconnected set D in okay an algebraically closed entire ultrametric box is especially defined through the round filters and the monotonous filters on D, in particular the T-filters: zeros of the weather, Mittag-Leffler sequence, factorization, Motzkin factorization, greatest precept, injectivity, algebraic houses of the algebra of the analytic components on D, difficulties of analytic extension. this can be utilized to the differential equation y'=hy (y,h analytic parts on D), analytic interpolation, p-adic team duality on meromorphic items and to the p-adic Fourier rework

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**Extra info for Analytic elements in p-adic analysis**

**Sample text**

Let r G]0,+oo[. j in ]0,-foo[ such that lim^i = 0. For every i G / , let a{ G E{ satisfy r — a < \a{\ < r and let a — (ai)i£i. Of course, a belongs to 7£, and satisfies lim|a;|i — r, hence r belongs to \S\. This shows that the valuation group of S is equal to IR. Finally we suppose that each field Ei is algebraically closed. Let P(x) q y \nxn = G S[x], with Xq = 1, and q > 0. We will show that P admits at least one n=0 zero in S. , # — 1, let (aj > n )j € j G 7£ satisfy V , ((at f n)ie/) = ^n? ,#.

L ). The m m , will be called (m,z) i* i) 1 < l < 5 ( m ). piercing of the sequence (T mi i) 1 <,-< m€lN s(m) • If a monotonous holes sequence ~m€lN has a piercing p > 0, it will be said to be well ; "rr J. cij. T - • a piercing i__. _ p > n 0, •it will i be said J. to be i__i__ If • a monotonous holes sequence has well pierced. If a monotonous filter T is run by a well pierced monotonous holes sequence, T will be said to be well pierced. *

We will show that cp is an ultrametric absolute value on ft. Clearly we have ip(a) ^ 0 whenever a G ft. 1 it is easily seen that we have