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 Ufimsk. Mat. Zh., 2012, Volume 4, Issue 4, Pages 38–44 (Mi ufa166)

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

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 38--44 \mathnet{http://mi.mathnet.ru/ufa166}