49 problems

Generalized Lax conjecture. All hyperbolicity cones are spectrahedral.

Lax conjecture. The curve $C$ is hyperbolic with respect to $p$ if and only if it has a determinantal representation by a symmetric matrix $M$ of linear forms such that $M(p)$ is a…

Let $h\in\mathbb{R}[x_1,\dots,x_n]$ be hyperbolic with respect to $\mathbf{e}=(e_1,\dots,e_n)\in\mathbb{R}^n$, and let $\Lambda_{++}(h,\mathbf{e})$ be its open hyperbolicity cone.…

Nonnegativity conjecture for Riesz kernels. For $\alpha\gg 0$, the Riesz kernel $q_\alpha(\sigma)$ takes nonnegative values on $K$.

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

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…

The optimality conjecture. The particular polynomial $S_{n,n-m}$ is optimal in the sense that

Common-orbit conjecture. If $P_{1},P_{2}\in\mathcal{R}_{n}$, then there exist differential operators $\Lambda_{1},\Lambda_{2}\in\mathcal{S}$ such that

Complex spectral-order conjecture. If $P\in\mathcal{C}_{n}$, then

Le's conjecture. Then $\operatorname{diag}(A)\in\Gamma(P)$ and

Certifying determinantal representation conjecture. There exists a polynomial $q(w_1,\ldots,w_{27})$ such that $pq$ has a certifying determinantal representation: there exist…

Let $S$ be a set of compositions of $n$ into at most $s$ parts, and let $\mathcal{L}(S)$ be the associated potential hyperbolic poset obtained by taking pairwise joins and then the…

Let $F$ be a real-rooted polynomial, let $\mathcal{L}_s(F)$ be the lattice associated with its hyperbolic slice, and let $\mathcal{L}^\Delta_s(F)$ denote its dual; write…

Let $F$ be a real-rooted polynomial and let $\mathcal{H}_s(F)$ be its hyperbolic slice, stratified by root multiplicity compositions; the associated hyperbolic poset records the in…

Monotonicity conjecture. The sequence $r(d)$ is decreasing.

Let $\Gamma_{n,d}$ be the class of symmetric degree-$d$ polynomials in $n$ variables. For a symmetric hyperbolic polynomial $p$, let its associated operator be the operator defined…

Let $\Gamma_{n,d}$ denote the class of symmetric degree-$d$ polynomials in $n$ variables, and let $\Delta_{\vec{1},\vec{1}}p$ be the indicated second directional derivative. A poly…

Let $T:\mathbb{R}[t]_{n,0}\to\mathbb{R}[t]_{d,0}$ be a diagonal linear map. A 0-sum hyperbolicity preserver is a map preserving the relevant 0-sum hyperbolicity property. Polya–Sch…

Generalized Lax conjecture. Every hyperbolic program is equivalent to a semidefinite program.

Extremal permanent conjecture. For all $M\in S(n,k)$ such that $T(M)=1$,

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…

Let $P$ be any degree $k$ lpm-polynomial, and let $\lambda,\mu\in\mathbb{R}^k$ with $\lambda$ majorizing $\mu$. Hyperbolic Horn conjecture. There exists a symmetric matrix $X$ such…

Let $P$) be a homogeneous PSD-stable lpm-polynomial. For polynomials $p$ and $q$ of the same degree, hyperbolic in direction $v$, say that $p$ majorizes $q$ in direction $v$ when,…

Hadamard-type inequality for hyperbolic polynomials. The claim extends the diagonal-versus-eigenvalue inequality from elementary symmetric polynomials to homogeneous, real, symmetr…

Let $H(e^n_k)$ denote the hyperbolicity cone of the elementary symmetric polynomial $e^n_k$, and let $DG(n,k)D$ be a diagonal congruence of the matrix $G(n,k)$ by a diagonal matrix…