Shidlovskii irreducibility (Q1842549)

Consider an \(n\times n\)-system of linear differential equations \(dY/dz= AY\) with \(A\in M_n (\mathbb{C} (z))\) and \(Y= (y_1, \dots, y_n)^t\) the vector of unknown locally complex analytic functions. In proving transcendence of the values of solutions at algebraic points Shidlovskij introduced the concept of (Shidlovskij) irreducibility. The system is said to be linearly Shidlovskij irreducible if the non-zero components of any solution vector are \(\mathbb{C}(z)\)-linear independent. In this paper the author gives a criterion for such irreducibility in terms of the structure of the differential Galois group of the system. This extends earlier results by Beukers, Brownawell, Heckman on the characterisation of Siegel normality.