4 problems

Let $F\in C^\infty(\overline{\Omega_r})$ be a function whose restriction to each complete circle $C_s:=\{x:|x-\gamma(s)|=r\}$ is the restriction of a polynomial in two variables of…

Let $H_1$ and $I_2$ be the first integrals in case 2 of Theorem, let $d=-(2p-1)/2$ with $p\in\mathbb{N}$, and let $\widetilde{H}_{3,p}$ denote the additional first integral. Polyno…

Explicit polynomial integral conjecture. The extra polynomial first integral for this system is given, up to the factor $(-1)^n$, by

The generalized Mathieu conjecture. The subspace $\mathcal M_B(\sigma)$ is a Mathieu subspace of $\mathbb C[z]$. The source presents this as a variation of Mathieu's conjecture and…