About the \(p\)-adic Yosida equation inside a disk (Q967561)

The paper under review deals with the \(p\)-adic Yoshida equation. Let \(K\) be a complete ultrametric algebraically closed field. Let \(d(0,r)=\{x\in K:|x| \leq r\}\) and \(d(0,r^-)=\{x\in K:~|x| < r\}\). Let \(\mathcal{A}\big(d(0,r^-)\big)\)~ be the \(K\)-algebra of holomorphic function defined in \(d(0,r^-)\), and let \(\mathcal{M}\big(d(0,r^-)\big)\) be the field of meromorphic functions in \(d(0,r^-)\). We denote by \(\mathcal{A}_b\big(d(0,r^-)\big)\) the subalgebra of bounded holomorphic functions. Let \(\mathcal{A}_u\big(d(0,r^-)\big)=\mathcal{A}\big(d(0,r^-)\big) \setminus \mathcal{A}_b\big(d(0,r^-)\big)\). Let \(\mathcal{M}_b\big(d(0,r^-)\big)\) and \(\mathcal{M}_u\big(d(0,r^-)\big)\) be the fields of fractions of \(\mathcal{A}_b\big(d(0,r^-)\big)\) and \(\mathcal{A}_u\big(d(0,r^-)\big)\), respectively. In the first part of the paper, the author gives the following theorem: Let \(F(X)\) be a rational function in \(\mathcal{M}_b\big(d(0,r^-)\big)(X)\). Suppose that the equation \((y')^m=F(y)\) admits a solution \(f\in \mathcal{M}_u\big(d(0,r^-)\big)\). Then \(F(X)\) is a polynomial of degree at most \(2m\). Next, the author studies the case where \(F(X)\) is a polynomial in \(K[X]\), and gives more precise results.