5 problems

Let $q$ be a prime power. A polynomial $f(X)\in\mathbb{F}_q[X]$ is exceptional if it induces a bijection on $\mathbb{F}_{q^k}$ for infinitely many positive integers $k$. Carlitz co…

Simple-zero conjecture. For any partitions $\lambda$ and $\mu$, take $\alpha>-1$ and $\beta>m_1$ such that the stated conditions are satisfied. If $\lambda$ is an even partition, t…

Let an exceptional orthogonal polynomial family have weight … where $W_0(x)$ is a classical orthogonal-polynomial weight and $\eta(x)$ is a polynomial whose degree equals the numbe…

Let $H_{\lambda}(z)$ be the generalized Hermite polynomial associated with a partition $\lambda$. Veselov's conjecture. For any partition $\lambda$, the zeros of $H_{\lambda}(z)$ a…

m-step Darboux conjecture. Every $\mathrm{X}_m$ orthogonal polynomial system for any codimension $m$ can be obtained by applying a sequence of at most $m$ Darboux transformations t…