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On the poles of Picard potentials. (English. Russian original) Zbl 1218.34102
Trans. Mosc. Math. Soc. 2010, 241-250 (2010); translation from Tr. Mosk. Mat. O.-va. 71, 270-282 (2010).
The author studies the existence of a global meromorphic fundamental system of solutions for a system of two differential equations
\[ E_x = \biggl(\begin{matrix} a_{11}z \quad u(x)\\ v(x) \quad a_{22}z\end{matrix}\biggr) E \tag{1} \]
where \(a_{11}\) and \(a_{22}\) are distinct complex numbers. Following Gesztesy and Weikard (1998) a pair \(u(x), v (x)\) of meromorphic functions on \(\mathbb C\) can be defined a Picard potential if, for each \(z\in \mathbb C\), the system (1) has a fundamental system of solutions which is meromorphic on the entire complex plane \(\mathbb C\) of the variable \(x\).
In Theorem 1 of this paper, the author obtains a necessary condition for an arbitrary pair \(u(x), v (x)\) of meromorphic functions to be a Picard potential, provided that at least one of them has a pole \(x_0 \in \mathbb C\): the function \(u(x), v (x)\) must have, at \(x_0\), a pole of order two with Laurent coefficient of special form. This necessary condition is rather restrictive, but it is still far from being sufficient, which is illustrated by Theorem 2, giving necessary and sufficient condition for the Picard property of monomial functions \(u(x),v (x)\).

34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
Full Text: DOI
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