21 problems

Gedeon–Proudfoot–Young’s conjecture. For a non-degenerate matroid $M$, the roots of $P_M(t)$ and those of the contraction $M/e$ satisfy interlacing properties. The source does not…

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

Let $\mathcal{H}_d$ denote the class of stable homogeneous polynomials of degree $d$, and let $\mathcal{H}_d/\{y^2\}$ denote the corresponding initial segment modulo $y^2$. The sta…

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

Let $A_i,B_i$ be positive definite matrices, let $g_i\in\mathbf{P}^{\mathrm{pos}}$, and let $f(x,y_1,\dots,y_e)\in\mathbf{P}_{1+e}$. The strong BMV conjecture. The determinant … ha…

Let $D_1,\dots,D_d$ be positive definite $\mu\times\mu$ matrices and set … The adjugate matrix $\operatorname{Adj}[M]$ is formed from the signed minors of $M$. The adjugate hyperbo…

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…

Let $P$ be a Cohen–Macaulay poset, with Chow polynomial $\operatorname{H}_{P}(x)$ and augmented Chow polynomial $\operatorname{H}^{\operatorname{aug}}_{P}(x)$. Interlacing conjectu…

Athanasiadis–Kalampogia-Evangelinou's interlacing conjecture. The polynomial $h_L(t)$ is real-rooted and is interlaced by the Eulerian polynomial $A_n(t)$ for every geometric latti…

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…

Let $\{x_i\}$ be the primary Gauss–Legendre nodes and $\{\bar z_i\}$ the secondary intermediate nodes. Asymptotic interlacing conjecture. The primary nodes and secondary intermedia…

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 $A,B\in\mathbb{S}^n$, and set $C=A\otimes B+B\otimes A$. An eigenvector is symmetric or skew-symmetric according as it lies in the $+1$ or $-1$ eigenspace of the commutation ma…

Let $A,B\in\mathbb{S}^n$, and set $C=A\otimes B+B\otimes A$. Let $\lambda_1\geq\cdots\geq\lambda_s$ and $\beta_1\geq\cdots\geq\beta_t$ be the even and odd eigenvalues of $C$, respe…

Let $A,B\in\mathbb{S}^n$ be positive definite matrices, and let $T$ denote the commutation matrix. A vector $u$ is symmetric when $Tu=u$, and a vector $w$ is skew-symmetric when…

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…

Let $P$ belong to $\{A_l\ (l\geq 1), B_l\ (l\geq 1), D_l\ (l\geq 4), E_l\ (l=6,7,8), F_4, G_2, H_3, H_4, I_2(p)\ (p\geq 3)\}$, and let $l=\operatorname{rank}(P)$. Write the zeros o…

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…

Interlacing conjecture. These distances interlace with those of order $3k+2$: