Local classification of linear ordinary differential equations. (Q1432529)

This paper is devoted to the problem of local classification of \(n\)th-order linear ordinary differential equations, \(n\geq 3\), \[ y^{(n)}= a_{n-3}(x) y^{(n-3)}+ a_{n-4}(x) y^{(n-4)}+\cdots+ a_0(x) y, \] with respect to contact transformations. At the end of 19th century, G.-H. Halphen obtained first results in this direction. In this work, the problem under consideration is reduced to the classification of section's germs of the linear ODE bundle with respect to the group of projective transformations of the base of this bundle. In this way, the problem is solved in a neighborhood of a regular point. The case \(n= 3\) has been studied in [\textit{V. A. Yumaguzhin}, Russ. J. Math. Phys. 4, 403--407 (1996; Zbl 0913.34033) and Differ. Geom. Appl. 6, 343--350 (1996; Zbl 0879.34009)].