Differential Galois theory and Lie symmetries (Q903691)

Let \(F\) be a differential field of functions meromorphic in a certain region \(U\) of the complex plane (for example, \(F=\mathbb C(z)\) or \(F=\mathbb C(\rho(z),\rho'(z))\)). The paper deals with a system of linear differential equations over \(F\) of the form \[ \frac{dy}{dz}=A(z)y,A(z)\in \mathfrak g\mathfrak l(n,F).\tag{1} \] Let \(G\) be the Galois group of the system. Roughly speaking, the group \(G\) is determined by the set of elements (invariants) of the ring of \(F\) which remain fixed under the action that induced by it linear transformation of variables. However, there is no any general algorithm for finding such invariants. For this purpose, it is suggested in the paper to connect with system (1) a vector field \[ X=\frac{\partial}{\partial z}+\sum\limits_{i,j=1}^na_{ij}(z)y_j\frac{\partial}{\partial y_i} \] and a Lie algebra \(sym_X\) consisting of vertical infinitesimal symmetries (=analytic vector fields) \(Y\) of the form \(Y=\sum\limits_{j=1}^nf_j(z,y)\frac{\partial}{\partial y_j}\) commuting with \(X\) in their common domain of definition. It is proved that the symmetry of the form \[ Y=\sum_{j=1}^np_j(y) {\partial \over\partial y_j},\quad p_j(y) \in F[y_1, \dots,y_n] \] that's what we need. For example, the system \[ \frac{dy_1}{dz}=ay_1+by_2,\frac{dy_2}{dz}=ay_2\quad (a,b\in F) \] has symmetry \(Y=y_1^2\frac{\partial}{\partial y_2}\). Whence one can conclude that the Galois group of this system is isomorphic to some subgroup of the matrix group \[ \left\{\left(\begin{smallmatrix} \lambda^2 & \mu \\ 0 & \lambda \\ \end{smallmatrix}\right):\lambda \in\mathbb C^\ast,\mu \in \mathbb C \right\}. \]