10 problems

The Faà di Bruno real-rootedness conjecture. For every $\alpha\in\mathbb{R}$ and $m=0,1,\dots$, $F_m(\alpha,y)$ has only real roots and

The composition conjecture. Every coefficient $f_i$ belongs to $\overline{\mathbf{P}}_d$, and

The scalar strong BMV coefficient conjecture. Every $f_i$ belongs to $\mathbf{P}^{\mathrm{pos}}$, and consecutive coefficients satisfy

Johnson–Bapat interlacing conjecture. If $A$ is positive semidefinite and $\eta(zA,-B)$ is not identically zero, then, for every $1\leq j\leq n$, the zeros of

Let $\mathcal{L}$ be a geometric lattice, let $a$ be an atom of $\mathcal{L}$, and let $\underline{\mathrm{H}}_{[a,\widehat{1}]}$ and $\mathrm{H}_{[a,\widehat{1}]}$ denote the augm…

For integers $q>0$ and $m=2,3$, let $H_n^{[m]}(q,t)$ denote the $(q,t)$-Hoggatt sums. The Sturm-sequence conjecture. The polynomial sequence … forms a Sturm sequence. The authors v…

Real-rootedness and Eulerian interlacing conjecture. The polynomial $h_{\mathcal L}(x)$ has only real roots and is interlaced by the Eulerian polynomial $A_n(x)$ for every geometri…

Let $M$ be a matroid and let $e$ be an element of its ground set. Let $M/e$ be the contraction of $M$ at $e$. A matroid is non-degenerate if its rank is $0$ or its Kazhdan–Lusztig…

Let $\{M_d\mid d\geq 0\}$ be a nice family of matroids, and let $Z_d(t)$ denote the $Z$-polynomial associated with $M_d$. For polynomials with real roots, say that $f(t)$ interlace…

Hyatt's conjectures. For $n\geq 1$, $B_n^+(x)$ interlaces $x^nB_n^+(1/x)$, and hence $B_n(x)=B_n^+(x)+x^nB_n^+(1/x)$ has only real zeros. For $n\geq 2$, $D_n^+(x)$ interlaces…