By Alain Escassut

ISBN-10: 9810222343

ISBN-13: 9789810222345

The behaviour of the analytic components on an infraconnected set D in ok an algebraically closed entire ultrametric box is especially defined by way of 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 homes 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 parts on D), analytic interpolation, p-adic team duality on meromorphic items and to the p-adic Fourier remodel

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**Example text**

V(P) = v(P,0) whenever P G L[x]. 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).

For every i G / , we put T,(x) = V^ ai >n :r n . g = 1 for every i G / , at least one of the zeros a, of Tz Ei satisfies \ai\q < |a«so|- But by hypothesis (ai ) o)ie/ belongs to 7£, hence does (ai)i£j. 1. Diarra) in so 0. of Let a = (a,i)iei £ 71. Then we have \\\a\\\ = ip(a). 7. , be defined as e t = 0 whenever i G J , and e,: = 1 whenever i ^ J. For convenience we put b = ae. Clearly b belongs to J, hence, we have |||a||| = inf \\a — t\\ < \\a — b\\. But now, we check that \\a — b\\ = sup |aj|j.

Besides, by hypothesis we have (1) deg{V) < deg(h) and (2) deg(W) < deg(g) Let B = Sg + Th-l. Then we have BP = (S0 + T0)gh + Vg + Wh- P. Since deg ( P ) < deg(g) + deg(h), by (1) and (2) we see that deg(Vg + VK/i - P ) < deg(g) + deg(h) and therefore F<7 + J ^ / i — P is just the remainder of the Euclidean division of BP by gh. 3, we have v(Vg + Wh — P ) > v(BP) = v(B) -f v ( P ) , and therefore by (1) and by definition of B it is seen that v(B) > 0. 4. 5: Let g,h G £/L[Z], &e monic and satisfy (7},h) = 1.

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