Differential Galois groups over Laurent series fields (Q2800460)

``Differential Galois theory studies linear homogeneous differential equations by means of their symmetry groups, the differential Galois groups. Such a group acts on the solution space of the equation under consideration, and this furnishes it with the structure of a linear algebraic group over the field of constants of the differential field. In analogy to the inverse problem in ordinary Galois theory, an answer to the question which linear algebraic groups occur as differential Galois groups over a given differential field provides information about the field and its extensions.''NEWLINENEWLINENEWLINEIf the field of constants is not algebraically closed (as in the paper under review) the linear algebraic groups are considered in the sense of group schemes rather than of point groups.NEWLINENEWLINENEWLINEIn the present paper, the authors solve the inverse problem of differential Galois theory over differential fields \(L\) whose field of constants is a Laurent series field \(K\) in characteristic zero and where \(L\) is finitely generated over \(K\). They show that every linear algebraic group over \(K\) occurs as differential Galois group of some differential extension of \(L\). The main result (Thm. 4.14) is deduced from the special case \(L=(K(x),\partial_x)\) of a rational function field over \(K\) by some argument which does not involve the particular shape of \(K\), and hence can also be used for other fields of constants (see Thm. 4.12 and Cor. 4.13).NEWLINENEWLINEFor proving the result for \(L=K(x)\) the authors apply the patching methods developed in an earlier work of the second and third author [Isr. J. Math. 176, 61--107 (2010, Zbl 1213.14052)] to realize split linear algebraic groups over larger fields of constants \(\tilde{K}\). Doing such realizations in a clever way -- for which they also introduce actions of finite groups on their differential patching data (cf. Def. 2.2) -- they can use Galois descent developed in [\textit{T. Dyckerhoff}, ``The inverse problem of differential Galois theory over the field \(\mathbb{R}(z)\)'', Preprint, \url{arXiv:0802.2897}] to obtain the realization of a given linear algebraic group \(G\) over \(K\) by descending the realization of its base change \(G_{\tilde{K}}\) for a suitable Galois extension \(\tilde{K}/K\).NEWLINENEWLINEThe paper is well organized and presented in a clear way. The authors also give the necessary prerequisites so that the reader can follow the paper without having to consult secondary literature.