18 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…

Let $f\in\mathbb{Z}[x]$ be a polynomial that induces a bijection on $\mathbb{Z}/p\mathbb{Z}$ for infinitely many primes $p$. Such a polynomial is called a Schur polynomial or excep…

Minimal-order recurrence conjecture. The minimal degree of an element of $\mathcal{S}$, and hence the minimal order of the recurrence relation for the exceptional Hermite polynomia…

Let $q$ be a prime power, and let $f(X)\in\mathbb{F}_q[X]$ be an exceptional polynomial, meaning that $f$ induces a bijection on $\mathbb{F}_{q^k}$ for infinitely many positive int…

Let $H_{\lambda}$ be the polynomial associated with a partition $\lambda$. Felix–Huybrechs–Vandebril conjecture. The nonzero zeros of $H_{\lambda}$ are always simple, including whe…

Let $\alpha>-1$, let $\lambda$ be an even partition, and let $\Omega_{\lambda,\mu}^{(\alpha)}$ be the polynomial whose zeros determine the exceptional zeros of…

A rational odd cyclic dressing chain is a cyclic chain of Darboux transformations with rational data and odd length, and an associated rational solution of the $A_{2n}$-Painlevé sy…

An exceptional polynomial system is a polynomial family associated with an exceptional differential operator, while a classical polynomial system is one of the classical orthogonal…

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…

Guralnick–Zieve conjecture. There are no examples of indecomposable exceptional polynomials of degree $p^r$, with $r>1$, other than those described in Propositions 8.4.15, 8.4.16,…

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…

Symmetry conjecture. The determinant is conjectured to satisfy

Minimal-order conjecture. The minimal order for recurrence relations of this form satisfied by the exceptional Hermite polynomials is $2w_F+1$, and it corresponds to the recurrence…

Admissibility characterization conjecture. The Wronskian determinant does not vanish on ${\mathbb R}$ if and only if $F$ is admissible.

Let $g^{(\alpha)}_{\mu}(z)$ be the function defined by the Wronskian construction of the rationally extended radial oscillator potential, with $\mu$ as specified by the multi-index…

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…

Darboux-Crum conjecture. Every exceptional orthogonal polynomial system is related to a classical orthogonal polynomial system by a Darboux-Crum transformation.

Let $\tilde{L}_{1,\nu+2}^{(\alpha)}(z)$ and $\tilde{L}_{2,\nu+2}^{(\alpha)}(z)$ denote the $X_{2}$-Laguerre polynomial systems of the first and second kinds, respectively. Let…