27 problems

Let $D_1,D_2,D_3$ be positive definite matrices, and let $M$ be the matrix whose entries are the coefficients of $y$ and $z$ in … A minor means the determinant of any finite square…

Let $f(x,y)=\sum f_i(x)y^i\in\mathbf{P}^{\mathrm{pos}}_2$, and form the upper-triangular coefficient matrix … A minor means the determinant of any finite square submatrix. The tota…

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…

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

Johnson's conjecture. If $A$ is positive semidefinite, then the polynomial $\eta(zA,-B)$ has all real zeros.

Let $\mu$ be a Hurwitz-stable distribution supported on one parity class. Consider the two-step parity flip--repair graph and its walk on each connected component: the walk flips a…

Let $\mu$ be a probability distribution on $\{0,1\}^n$ with generating polynomial … Assume that $g_\mu$ has nonnegative coefficients and is Hurwitz stable, meaning that…

For each $n\in\mathbb{N}$, let $X_n^\star$ be the idealizer associated with quasi-stable polynomials, and let $Y_n^\star$ be the corresponding family defined by $Y_n^\star=Y_n$ for…

For each $n\in\mathbb{N}$, let $X_n\subset\mathbb{R}_n^+$ be the largest subsemigroup containing $\mathcal{H}_n$ and $\mathbb{I}_n$ such that, for every $m\leq n$, Hadamard multipl…

Shortest-path contact-order conjecture. If $t=0$, then $p_A$ has contact order $2\ell$ at $(-1,1)$, and if $t\in(0,1)$, then $p_A^t$ has contact order $2\ell+2$ at $(-1,-1)$.

A matroid has the weak half-plane property when its set of bases agrees with the support of a stable polynomial. A relaxation of a matroid is obtained by relaxing a circuit-hyperpl…

Brändén–Huh's conjecture. The following conditions are equivalent:

Diagonal-monomial reachability conjecture. For any monomial $Z^\beta$ appearing in a psd-stable polynomial, there is a diagonal monomial $Z^\alpha$ appearing in $f$ which can be re…

Let $d$ be a Dyck path, let $G(d)$ be its indifference graph, and let $X_{G(d)}$ be its chromatic symmetric function. For any finite number of variables, restrict $X_{G(d)}$ to tho…

A homogeneous multiaffine polynomial is a polynomial in which every variable has degree at most one and all monomials have the same total degree. Multiaffine Schur–Horn conjecture.…

Let $D$ be a positive definite diagonal matrix, and let $p(x)=e_k(Dx)$. Diagonal-rescaled elementary symmetric conjecture. The polynomial $p(x)$ has the spectral containment proper…

A homogeneous multiaffine stable polynomial is a stable polynomial that is homogeneous and multiaffine. Quadratic spectral containment conjecture. Every quadratic homogeneous multi…

A homogeneous multiaffine stable polynomial $p\in\mathbb{R}[x_1,\ldots,x_n]$ has the spectral containment property if, for every symmetric matrix $A\in H(P)$, some eigenvalue vecto…

Full numerator criterion. For every $f\in\mathbb{C}[z_1,z_2]$, $f/p$ is locally $H^{\infty}$ if and only if $f\in\mathcal{I}$.

Boundary-order criterion. One might conjecture that $q/p$ is bounded on $\mathbb{H}^2$ whenever $q$ vanishes to at least the same order as $p$ at every boundary zero of $p$.

For each integer $n\geq4$, let $V_{2n}$ denote the generalized Vámos matroid in the infinite family studied by Burton, Vinzant and Youm. A matroid has the half-plane property when…

Let $A=(a_{ij})$ be an $n\times n$ real matrix whose entries are weakly decreasing down each column, so $a_{i,j}\geq a_{i+1,j}$ for $1\leq i\leq n-1$ and $1\leq j\leq n$. Let $J_n$…

Let $G=(V,E)$ be a graph and let $0<q<1$. Consider the associated random cluster measure and its multiaffine probability-generating polynomial. Random-cluster Rayleigh conjecture.…

For $n\geq 2$, let $\widetilde{B}_n(\mathbold{x},\mathbold{y})$ be the multivariate refinement defined by … Write $\mathfrak{S}_{\mathbb{R}}[\mathbold{x},\mathbold{y}]$ for the cla…

Let $A=(a_{ij})$ be an $n$-by-$n$ monotone column matrix, meaning that its real entries satisfy $a_{ij}geq a_{i+1,j}$ for all $1\leq i\leq n-1$ and $1\leq j\leq n$. Let $J_n$ be th…