Summary
Let K be an algebraically closed complete ultrametric field. A bounded closed subset D of K is said to be calibrated if its diameter and the diameters of its holes lie in ¦K¦. It is said to be strongly infraconnected if for every hole of center a, of diameter r, there is sequence an in D such that ¦an−a¦=¦am−an¦=r whenever n≠m. Let H(D) be the Banach algebra of the analytic elements in D. Each element f ε H(D) reaches its maximum at a point α ε D if and only if D is strongly infraconnected and calibrated. Let D be open closed strongly infraconnected and let\(F_x (Y) = \sum\limits_{s = 0}^\infty {\xi _s (x)} Y^r \) be a Taylor series with coefficientsH(D). Assume all the ξs are quasi-invertible and there exist n ε N such that
whenever l=0, ...,n−1. Let M be the set of the αεD such that ξ0(α)=ξ1(α)=...= =ξn−1(α)=0. Assume there is a bounded function r from DM intoR + such that Fx has exactly n zeroes (distinct or not) in the disk δx:¦Y¦⩽r(x). Then Fx admits a kind of Hensel factorization in H(D)[Y] in the form Px Gx where Px is the degree n monic polynomial (with coefficients in H(D)) the zeroes of which are the n zeroes of Fx in δx.
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Escassut, A. Maximum principle for analytic elements and Lubin-Hensel's theorem inH(D[Y] . Annali di Matematica pura ed applicata 135, 265–278 (1983). https://doi.org/10.1007/BF01781071
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DOI: https://doi.org/10.1007/BF01781071