Download e-book for iPad: Analytic elements in p-adic analysis by Alain Escassut

By Alain Escassut

ISBN-10: 9810222343

ISBN-13: 9789810222345

The behaviour of the analytic parts on an infraconnected set D in ok an algebraically closed whole ultrametric box is principally defined through the round filters and the monotonous filters on D, in particular 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 is often utilized to the differential equation y'=hy (y,h analytic parts on D), analytic interpolation, p-adic workforce duality on meromorphic items and to the p-adic Fourier remodel

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So we have shown that = tpjr. We now suppose that there does not exist a G F such that r — ip(x — a). There exists an G F such that r < i^>(x — an) < r -\ . Let pn = ip(x — an). - I <^ in/.. V»p»r if T*_ i« tViP> n r r n l a r filter nf r p n f p r rv nnH i/>(x — an) < \an — b\. Further, if Tn is the circular filter of center an and diameter pn we have (3) ij)(x — b) = \b — an\ = ¥Tn{x — b).

E v{ 1 i-Q) v(Q) + Hence by (1) we obtain (8) v(Vg + Wh-Q) > v(Q) + A(#, h). Wh-Q) By (1) , (5) it is seen that W)) > v(Q). We \ will check (9) vyvv v(W) ^_ v\\^). we (10) v(V) > v(Q). Indeed v(V) > v(Q). Inde we have v(h) = 0, hence by (2) we see (11) v{£h) > v(Q) by (11) we obtain (10). Finally, by definition we have deg(£) = deg(Q) — de> deg(gh) and therefore Jf>n(Ph\\ < max (deg(h), , deg(Q) - dtg(g (12) deg(V) < max {deg{Vx\ I deg(£h)) < deg(g)).

Next, the valuation group of Q,p is easily seen to be isomorphic to (flj, + ) . For convenience, we take u; = p. 10 Q,p has a completion (Dp that is algebraically closed. The valuation group of (Dp is then isomorphic to (Q,+) like this of £lp. 6, the residue class field of O p is an algebraic closure of Wp and the one of (Dp is seen to be the same. We will now study the p3-th roots of 1. We will need certain technical lemmas. 4: Let s G IN*. For every n £ IN* such that n < p9, we have KO l n ) < ——— < - .

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Analytic elements in p-adic analysis by Alain Escassut


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