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 whole ultrametric box is especially defined by means of 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 components 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 crew duality on meromorphic items and to the p-adic Fourier remodel

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F. F. For For every fi < v(a — b), w we have va(h,ii) — VJ,(/I,/X) . n n For For + For every even/ IR, 7V+(P,/i) IR,7V+ il ]] •. For every //i G IR R,JV (P,/ij + + (P, /i) — every // G IR, 7V IR, N~(P,fji) is equal to the number i=o of zeros admitted by P in the circle C(0,u? ) number of zeros by P in the C(0,u? - /on ) in Jf. F. A (resv. circle continuous in (resp. iv admitted ^ + in F. ) (resp. )) is decreasing and continuous on the rii £/&e /e/£ (resp. oni the right). the left (resp. on the right).

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. We will check that ||PQ|| = ||P||||Q|| whenever P,Q e B.

Monotonous and circular filters Proof: 23 We may assume b = 0 and then the proof is immediate. 5: Let a e L,rur2 e]0,+oo[ with \a - a\ < n < r 2 . ,r2)) = r ( 6 , i - , l ) . 6: Let T be the increasing (resp. decreasing) filter of center a and diameter R > \a — a\, on L\ {a}. Then ^{T) is the decreasing (resp. increasing) filter of center b and diameter —. 7: Let a e L be such that \a - a\ ^ r. Let a e L be such that \a - a\ ^ r. 7(C(a,r)) Proof: Proof: Then Then = c(7(a),]^jj). When x belongs to C ( a , r ) we have When x belongs to C ( a , r ) we have \a — a\2 7(z) - 7 ( a ) = (x — a)(a — a)' hence 7(C(oj,r)) C C[j(a), -——-).

### Analytic elements in p-adic analysis by Alain Escassut

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