Wronskien et équations différentielles $p$-adiques

Jean-Paul Bézivin¹

¹ 1, Allée Edouard Quincey 94200, Ivry-sur-Seine, France

Acta Arithmetica, Tome 158 (2013) no. 1, pp. 61-78.

Résumé

We prove an inequality linking the growth of a generalized Wronskian of $m$ $p$-adic power series to the growth of the ordinary Wronskian of these $m$ power series. A consequence is that if the Wronskian of $m$ entire $p$-adic functions is a non-zero polynomial, then all these functions are polynomials. As an application, we prove that if a linear differential equation with coefficients in ${\mathbb{C}}_p[x]$ has a complete system of solutions meromorphic in all ${\mathbb{C}}_p$, then all the solutions of the differential equation are rational functions. This is also the case when the linear differential equation has coefficients in $\mathbb{Q}[x]$, and has, for an infinity of prime numbers $p$, a complete system of meromorphic solutions in a disc of ${\mathbb{C}}_p$ with radius strictly greater than $1$.

Détail

DOI : 10.4064/aa158-1-4

Mots-clés : prove inequality linking growth generalized wronskian p adic power series growth ordinary wronskian these power series consequence wronskian entire p adic functions non zero polynomial these functions polynomials application prove linear differential equation coefficients mathbb has complete system solutions meromorphic mathbb solutions differential equation rational functions linear differential equation has coefficients mathbb has infinity prime numbers complete system meromorphic solutions disc mathbb radius strictly greater