The p-adic differential equation y' = wy in the closed unit disk

• Autores: Alain Escassut, Marie-Claude Sarmant
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 41, Fasc. 2, 1990, págs. 165-174
• Idioma: inglés
• Títulos paralelos:
  • La ecuación diferencial p-ádica y' = wy en el disco unidad cerrado
• Enlaces
  • Texto completo
• Resumen

  • Let $\mathbb{K}$ be a complete ultrametric algebraically closed field. When $D$ is a set in $\mathbb{K}$, we denote by $H(D)(resp. H_b(D))$ the set of the analytic elements in $D$ (resp. the bounded analytic elements in $D$) . Let $\mathcal{B} = (b_n)_{n\in\mathbb{N}}$ be a sequence in $\mathbb{K}$ such that $1 <\vert b_{n+1}\vert < \vert b_n\vert$ and $\lim_{n \to \infty}\vert b_n\vert = 1$ and let $$\Lambda_\rho(\mathcal{B})=\mathbb{K}\setminus\big(\cup^\infty_{n=1}d(b_n,\rho^-)\big)$$ First we characterize the elements $f\in\cap_{\rho>0}H_b(\Lambda_\rho(\mathcal{B}))$ which are Meromorphic Products [$S_2$] in the form $\prod^\infty_{n=1}(x-a_n/x-b_n)$. Next , we translate the results of analytic extension through a $T$-filter [$S_3$] into terms of Meromorphic Products. We then apply that to the extension of a differential equation $y'=\omega y$ defined in the disk $d(0,1)(\vert x\vert\leq 1)$ to $K\setminus\mathcal{B}$; $y$ has an extension in the form $\lambda\prod^\infty_{n=1}(x-a_n/x-b_n)$ while $\omega$ has the extension $\sum^\infty_{n=1}((1/x-a_n-1/x-b_n))$.\newline As a consequence we can characterize the $\omega\in H(d(0,1))$ such that the equation $y'=\omega y$ has solutions in $H(d(0,1)$: these $\omega$ are the series $\sum^\infty_{n=1}((1/x-a_n)-(1/x-b_n))$. When $\mathbb{K}=\mathbb{C}_p$ such a series may be equal to a constant $\lambda\in\mathbb{C}_p$ in $d(0,1)$ if and only if $\vert\lambda\vert < \rho^{-1/p-1}$.