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 ℂₚ[x] has a complete system of solutions meromorphic in all ℂₚ, then all the solutions of the differential equation are rational functions. This is also the case when the linear differential equation has coefficients in ℚ[x], and has, for an infinity of prime numbers p, a complete system of meromorphic solutions in a disc of ℂₚ with radius strictly greater than 1.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-1-4,
author = {Jean-Paul B\'ezivin},
title = {Wronskien et \'equations diff\'erentielles p-adiques},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {61-78},
zbl = {1278.12004},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-1-4}
}
Jean-Paul Bézivin. Wronskien et équations différentielles p-adiques. Acta Arithmetica, Tome 161 (2013) pp. 61-78. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa158-1-4/