# Analytic elements in p-adic analysis by Alain Escassut

By Alain Escassut

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

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This finishes showing that the mapping T —► ipy is injective. Now we will show that this mapping is also surjective. Indeed let ip be an absolute value on F(x), and let r — inf ip(x — A). A£F We first suppose that there exists a G F such that ip(x — a) = r . Since tp is an absolute value, we check that r > 0 because if r — 0, we have 4>{h) = h(a) for every h G F(x) and then %j) is not an absolute value. Hence we can assume 4. Ultrametric absolute values and valuation functions... 33 r > (0. Let T be the circular filter of center a, of diameter r.

2 we have |P 1 + Z (0)| < 1, and therefore <^(1 + z) < 1. Now, the ultrametric inequality will be easily deduced. Let a, 6 G ft satisfy 0 < \a\ < \b\. We have y>(a-f b) =

(b). Thus we have now proven y> to be an ultrametric absolute value that extends the one of L. 3, this absolute value on ft is unique. This ends the proof. 4: Let L be complete and let ft be an algebraic closure of L provided with the unique absolute value \ .

We put R = \o>n — dm\ for Ti ^ m and a = ao- The circular filter of center a of diameter R on 26 Analytic elements in p-adic analysis L is clearly secant with D because each set A n = r ( a n , r ' , r " ) with r' < R < r" belongs to its canonical generating system and contains cifn for every TYI ]> n hence its intersection with D is a circular filter C on D less thin than the sequence (an)neiN- That ends the proof. Notations: Let V be an extension of L provided with an absolute value that extends the one of L.