Nonlinear Conditions for Ultradifferentiability.

A remarkable theorem of Joris states that a function f is $C^{\infty }$ if two relatively prime powers of f are $C^{\infty }$ . Recently, Thilliez showed that an analogous theorem holds in Denjoy-Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris's result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.