Article
Ufa Mathematical Journal
Volume 4, Number 4, pp. 38-43
On automorphic systems of differential equations and $\mathrm{GL}_2(\mathbb{C})$-orbits of binary forms
Bibikov P.V.
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In the work we suggest a new approach to study the classical algebraic problem on classifying $\mathrm{GL}_2(\mathbb{C})$-orbits of binary forms be means of differential equations. We construct and study an automorphic system of differential equations $\mathcal{S}$ of at most fourth order whose space of solutions is the $\mathrm{GL}_2(\mathbb{C})$-orbit of a given binary form $f$. In the case when system $\mathcal{S}$ is of order $2$ or $3$ it can be explicitly integrated. In the most difficult case of a fourth order system $\mathcal{S}$ it is shown that the system can be reduced to a first order differential equation of Abel kind and a linear partial differential equation of the first order.