By Alain Escassut

The behaviour of the analytic parts on an infraconnected set D in ok an algebraically closed whole ultrametric box is principally defined by means of the round filters and the monotonous filters on D, specifically the T-filters: zeros of the weather, Mittag-Leffler sequence, factorization, Motzkin factorization, greatest precept, injectivity, algebraic homes 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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**Example text**

Actually all the properties of the Analytic Elements are depending on this. At the same time we will define the valuation functions v(h^fi) that are convenient mainly because they are piecewise linear. They just translate the properties of the abolute values on F(x) into terms of piecewise linear functions. ,xq] be an algebra of polynomials in q indeterminates, with co efficients in L. ,», ^i* x q we i Pu^ xJq<. tiql || . || is a multiplicative norm of L-algebra. , xq]. \\ . || is clearly seen to be an ultrametric norm of L-vector space on B.

11: Let P(x) P(x) = Y^ Y^aajxjJxi Let e F xx 4. Ultrametric absolute values and valuation functions... 31 ■J(P,. ) and — j — has a left side derivative (resp. a right side derivative) equal to N+(P, fi) (res' (resp. (P,fi) N-(P,»)). f: It is easily seen seei that the equality (1) iV+(P,^) = N~(P,fi) Proof: P,fi) holds for fj, but finitely many mai values, at most n. ) are decreasing. By continuity we see thatit the function i« continuous ronfiniimiQ < 7V + (P,3, //) is continuous on the left at each point while N~(P,fi) i is • i_x on the_ _right at each point.

8: For every s £ IN, G3 is an Eisenstein Proof : First we suppose 5 = 1. We have p-l h=0 h j=0 polynomial. 4, we have I . 1 x*. Besides by Lemma ( . *_ j=o ^ ' . 1 = - for every j = 0 , . . ,p — 2 and therefore Gi is an Eisenstein polynomial. Now we consider the general case s G IN*. First we put T3(x) = (l+x)p . 4, it is seen that Ti(x) is of the form l + xp + Ti(x) with TI(X) G xp7Lp[x] and deg(ri) = p — 1. Then by an immediate induction, we see that Ts(x) is of the form 1 + xp' -f r3(x) with r3(x) G xp7Lp[x] and deg(r3) = p9 — 1.