Hilbert's Twenty- First Problem
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, was phrased like this (English translation from 1902).In fact it is more appropriate to speak not about differential equations but about linear systems of differential equations: in order to realise any monodromy by a differential equation one has to admit, in general, the presence of additional apparent singularities, i.e. singularities with trivial local...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, was phrased like this (English translation from 1902).In fact it is more appropriate to speak not about differential equations but about linear systems of differential equations: in order to realise any monodromy by a differential equation one has to admit, in general, the presence of additional apparent singularities, i.e. singularities with trivial local monodromy. In more modern language, the (systems of) differential equations in question are those defined in the complex plane, less a few points, and with a regular singularity at those. A more strict version of the problem requires these singularities to be Fuchsian, i.e. poles of first order (logarithmic poles). A monodromy group is prescribed, by means of a finite-dimensional complex representation of the fundamental group of the complement in the Riemann sphere of those points, plus the point at infinity, up to equivalence.
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