14 problems

Veselov's conjecture. This Wronskian has simple zeros, except possibly at $x=0$.

Let $\textbf{\textsl{k}}=(k_1,\ldots,k_l)\in\mathbb{N}^l$ and let $Q_n^{\textbf{\textsl{k}}}(x,y)$ be the associated polynomials. Integrality and divisibility conjecture. The polyn…

Let $k\in\mathbb{N}$ and let $\textbf{\textsl{k}}=[k,k+1]$ be the Krein–Adler sequence, with $l=2$. For the polynomial $Q_{k+1}^{\textbf{\textsl{k}}}(x,y)$ and Hermite polynomials…

Let $l\geq 2$ and let $\textbf{\textsl{k}}=(k_1,\ldots,k_l)\in\mathbb{N}^l$ be strictly increasing. Write $\tilde{\textbf{\textsl{k}}}=(k_1,\ldots,k_{l-1})$. For the polynomials…

Conjecture 1. The identities

Zero-distribution conjecture. For sufficiently large $n$, if $n-K$ is even, then $q_n$ has $(n-K)/2$ negative zeros and $(n-K)/2$ positive zeros. If $n-K$ is odd, then $q_n$ has at…

Let $\mu$ be a partition and let $V_\mu(x)$ be the associated Wronskian of Hermite polynomials. Hermite-Wronskian multiple-root conjecture. If $x=v$ is a multiple root of…

Let $\mathtt{H}_{\xi_1},\ldots,\mathtt{H}_{\xi_\ell}$ be Hermite polynomials, and let the associated partition be … Write $d_{\xi}$ for the quantity associated with the index seque…

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…

Hermite-polynomial upper-bound conjecture. For all $n\in\mathbb{N}^+$ and $x>0$,

For each $d\geq 1$, define the Hermite polynomial by … Let $\mathscr{R}_d$ denote the set of real polynomials of degree $d$ having only real periodic points. Hermite polynomial con…

Felder–Hemery–Veselov conjecture. For every doubled partition $\lambda=(\mu_1^2,\dots,\mu_n^2)$, $H_\lambda$ has no real roots and has as many imaginary roots as there are odd numb…

Let $\nu=(\mu_1^2,\dots,\mu_n^2)$ be a doubled partition with distinct parts, where $\mu_i^2$ denotes that the part $\mu_i$ occurs twice, and let $W_{\nu}(z)$ be its associated Wro…

Let $\lambda$ be a partition, and let $W_{\lambda}(z)$ denote the Wronskian of Hermite polynomials associated with $\lambda$. Simple-zero conjecture. For every partition $\lambda$,…