About the p-adic Yosida equation inside a disk

Abstract

Let K be a complete ultrametric algebraically closed field and let ℳ( d(0, R ℒ)) be the field of meromorphic functions inside the disk d(0, R −) = { x ∈ K ∣ ∣x∣ < R}. Let ℳ b ( d(0, R ℒ)) be the subfield of bounded meromorphic functions inside d(0, R −) and let ℳ u ( d(0, R ℒ)) = ℳ( d(0, R ℒ)) ∖ ℳ b ( d(0, R ℒ)) be the subset of unbounded meromorphic functions inside d(0, R −). Initially, we consider the Yosida Equation: ( d y d x ) m = F ( y ) , where m ∈ ℕ * and F(X) is a rational function of degree d with coefficients in ℳ b ( d(0, R ℒ)). We show that, if d ≥ 2 m + 1, this equation has no solution in ℳ u ( d(0, R ℒ)). 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 ℳ u ( d(0, R ℒ)). Particularly, the elliptic equation may not.