A real trigonometric M-polynomial of degree \(n\) is one with the maximal number \(2n\) of real critical points (on the circle). V. Arnold [in: The Arnold-Gelfand mathematical seminars: Geometry and singularity theory, 101-106 (1997; Zbl 0904.58007)] described the space of such polynomials as a convex cone formed by the mirror arrangement of the reflection group \(A_{2n-1}\) and calculated the number of topological types of generic M-polynomials (i.e., with all critical values distinct) of a given degree. The authors apply the same technique and obtain similar results for the spaces of trigonometric M-polynomials which are Möbius, i.e. satisfy \(g(\varphi+\pi)=-g(\varphi)\), or are odd, i.e. satisfy \(g(-\varphi)=-g(\varphi)\). These spaces are identified with cones formed by the mirror arrangement of the reflection groups \(B_n\). Numbers of topological types of Möbius and odd M-polynomials of a given degree are found.
For the entire collection see [Zbl 0890.00033].