Wronskien et équations différentielles $p$-adiques

Tom 158 / 2013

Jean-Paul Bézivin Acta Arithmetica 158 (2013), 61-78 MSC: rimary 12H25; Secondary 11S80. DOI: 10.4064/aa158-1-4


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 $1$.


  • Jean-Paul Bézivin1, Allée Edouard Quincey
    94200, Ivry-sur-Seine, France

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