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On the constructive inverse problem in differential Galois theory. (English) Zbl 1088.34075
The authors show how to construct differential equations over \(\mathbb{C}(x)\) with Galois group \(G\), where \(G^0\) is of the form \(G_1\cdots G_r\), all \(G_i\) being simple groups of type \(A_l\), \(C_l\), \(D_l\), \(E_6\) or \(E_7\). To this end, they give sufficient conditions a differential equation to have a given semisimple group as Galois group. They discuss criteria allowing one to reduce the inverse problem for arbitrary linear algebraic groups over \(\mathbb{C}(x)\) to find equivariant differential equations with given connected Galois groups over an arbitrary finite Galois extension \(K\) of \(\mathbb{C}(x)\).

MSC:
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
12H05 Differential algebra
12H20 Abstract differential equations
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