Mathematics

E.Wilczynski described in [1] all invariants of linear ODEs $y^{(n+1)}+p_{n}(x)y^{(n)}+\dots + p_0(x)y = 0$ considered up to the pseudogroup of transformations $(x,y)\mapsto (\lambda(x),\mu(x)y)$. These invariants can also be interpreted as differential invariants of non-parametrized curves in $n$-dimensional projective space. Using the universal linearization of ordinary differential equations, we show how these invariants can be generalized to become invariants of non-linear ODEs viewed up to contact transformations. Next, we discuss the role of these invariants in the canonical coframe for ordinary differential equations (see [2]). In particular, we show that if all these invariants vanish, then all other invariants of the canonical coframe become first integrals of the original equation. Further, using the technique developed in [3], we show how to constuct ODEs with vanishing Wilczynski invariants and present several non-trivial examples. References [1] E.J. Wilczynski, Projective differential geometry of curves and ruled surfaces, Leipzig, Teubner, 1905. [2] B. Doubrov, B. Komrakov, T. Morimoto, Equivalence of holonomic differential equations, Lobachevsky Journal of Mathematics, v.3, , pp.39-71. [3] R. Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory}, Proc. Symp.\Pure Math. 53 (1991), pp. 33-88.