About the p-adic Yosida equation inside a disk |
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Authors: | Abdelbaki Boutabaa |
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Institution: | Laboratoire de Mathématiques, UMR 6620, Université Blaise Pascal (Clermont-Ferrand), Les Cézeaux, 63177 Aubiere Cedex, France |
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Abstract: | Let K be a complete ultrametric algebraically closed field and let ?(d(0, R?)) be the field of meromorphic functions inside the disk d(0,R−) = {x ∈ K ∣ ∣x∣ < R}. Let ?b(d(0, R?)) be the subfield of bounded meromorphic functions inside d(0,R−) and let ?u(d(0, R?)) = ?(d(0, R?)) ? ?b(d(0, R?)) be the subset of unbounded meromorphic functions inside d(0,R−). Initially, we consider the Yosida Equation: , where m ∈ ?* and F(X) is a rational function of degree d with coefficients in ?b(d(0, R?)). We show that, if d ≥ 2m + 1, this equation has no solution in ?u(d(0, R?)).Next, we examine solutions of the above equation when F(X) is apolynomial with constant coefficients and show that it has no unbounded analytic functions in d(0,R−). Further, we list the only cases when the equation may eventually admit solutions in ?u(d(0, R?)). Particularly, the elliptic equation may not. |
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