13 problems

Let $\{P_n\}_{n=0}^\infty$ be an orthogonal polynomial system with respect to an arbitrary positive measure $d\mu=W\,dx$ supported on an interval $I$ of the real line. Assume that…

For integers $d$ and $0<k<m<d$, let $P_{k;d}(x)$ and $P_{m;d}(x)$ denote the truncated binomial polynomials used in the paper. Common-root conjecture. The polynomials $P_{k;d}(x)$…

Let $d\geqslant 1$ be the dimension of the base $\mathbb{R}^d$, let $k\geqslant 1$ be a differential order, and set … Let $W_d^k$ denote the Wronskian $N$-ary bracket on…

Let $T_r=A_{2r}/\mathbb{Z}_2$ be the tadpole Dynkin diagram, and let $\mathcal{T}_r(\mathbf{x}_r;q)$ denote its tadpole Nahm sum. The sum $\mathcal{T}_r(1,\ldots,1;q)$ is the princ…

General tadpole modularity conjecture. For even $n=2k\geq 2$,

Let $V\in\operatorname{Gr}_{k,n}(\mathbb{R})$. Regard the Wronskian $\operatorname{Wr}(V)$ as having a zero at $-\infty$ when its degree is less than $k(n-k)$. Wronskian positivity…

Let $V\in\operatorname{Gr}_{k,n}(\mathbb{R})$. Suppose that all complex zeros of the Wronskian $\operatorname{Wr}(V)$ are real, and let $I\subseteq\mathbb{R}$ be any interval on wh…

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

Let … be a linear ordinary homogeneous differential equation of order $n$ with real-valued continuous coefficients, disconjugate on an interval $I$. For a positive integer…

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 $\lambda$,…

Let $U\in\mathcal{G}_k\mathcal{P}_n$ admit a basis of top powers, and let $W(U)$ denote its Wronskian covariant. The notation $U^{(\ell-1)}$ denotes the associated derived subspace…