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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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##### References:
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