A note on \(p\)-adic linear differential equations (Q5931322)

Let \(\overline Q\) be the algebraic closure of \(Q\) in the field \(C_p\) of \(p\)-adic numbers, \(M_p\) be the field of meromorphic functions over \(C_p\), \(K\) be a field between \(\overline Q\) and \(C_p\) and \(F\) be a Picard-Vessiot extension of \(K(x)\). The author proved earlier [Lect. Notes Pure Appl. Math. 192, 49-59 (1997; Zbl 0942.12004)] if \(K=\overline Q\) then \(F\cap M_p=K(x)\). In the paper under review he shows that it is not true when \(K\neq\overline Q\).NEWLINENEWLINENEWLINEThe author's investigation is connected with the question of under what conditions is any meromorphic solution of a linear differential homogeneous equation over \(\overline Q(x)\) a rational function. Let us notice that for this purpose it is enough to have all characteristic determinants equal zero in singular points of the equation [see for details the reviewer, Math. Notes 33, 453-454 (1983); translation from Mat. Zametki 33, 881-884 (1983; Zbl 0549.34005)].