
On automorphic systems of differential equations and $\mathrm{GL}_2(\mathbb C)$orbits of binary forms
P. V. Bibikov^{} ^{} Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia
Abstract:
In the work we introduce a new method for studying the classical algebraic problem of classifying $\mathrm{GL}_2(\mathbb C)$orbits of binary forms with the help of differential equations. We construct and study an automorphic system of differential equations $\mathcal S$ of the fourth order, whose solution space coincides with the $\mathrm{GL}_2(\mathbb C)$orbit of a fixed binary form $f$. The system $\mathcal S$ is integrable in cases when it is of the second and third order. In the most difficult case, when the system is of the fourth order, we prove that the system $\mathcal S$ can be reduced to a first order differential equation of the Abel type and a linear partial differential equation of the first order.
Keywords:
binary forms, jet space, differential invariants, automorphic differential equations.
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UDC:
517.957+512.745 Received: 30.09.2011
Citation:
P. V. Bibikov, “On automorphic systems of differential equations and $\mathrm{GL}_2(\mathbb C)$orbits of binary forms”, Ufimsk. Mat. Zh., 4:4 (2012), 38–44
Citation in format AMSBIB
\Bibitem{Bib12}
\by P.~V.~Bibikov
\paper On automorphic systems of differential equations and $\mathrm{GL}_2(\mathbb C)$orbits of binary forms
\jour Ufimsk. Mat. Zh.
\yr 2012
\vol 4
\issue 4
\pages 3844
\mathnet{http://mi.mathnet.ru/ufa166}
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