By Alain Escassut

The behaviour of the analytic parts on an infraconnected set D in okay an algebraically closed entire ultrametric box is principally defined through the round filters and the monotonous filters on D, specially 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 can be 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 remodel

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**Example text**

By (VV„), we have \\bn\\ < r o whenever n G IN, hence |/it|,- < ro for each i G / , and therefore (hi)i^j belongs to 71. We put h = (hi)iei, and UJ — ip(h). We will show (2) |a; — an\ < r n _ ! whenever n G IN*. Let n G IN* be fixed. It is seen that for every m > n, we have ||6 n — 6 m || < r n _ i , hence for every i G I we have | 6 i n — 6ijTn|i < r n _!. Beside, since ( / m ) m > n makes a partition of Xn_|_i, for every i G -Xn+i there exists m > n such that i G I m , and then we have |6;>n - h{\{ = \biin - bit9^\i = |6 i>n - 6ijTn|i < r n _ !

Iei We will denote by ip the mapping from 71 into IR+ defined as <£>((a,),ej) = lim |a,L. Then u> is seen to be a multiplicative semi-norm of the L-algebra 7Z. u 71 We put J = Ker(

Therefore, P divides Q in L[x\. Let Q(x) — P(x)T(x). Since both P, Q are monic, so is T, hence we have v(P) < 0, v(T) < 0. But since v(Q) = 0, P, T must satisfy v(P) = v(T) = 0, and thefore |P(0)| < 1. Now we assume |P(0)| < 1. Suppose v(P) < 0. There exists b G ML such that v(bP) = 0, and then we have |&P(0)| < 1, hence 0 is a zero of bP. Besides we notice (1) deg(bP) < deg(P). Let bP = xd