34 problems

Let $F$ be a generic homogeneous form, and let $\mathcal{R}_F$ denote the factor of the resultant appearing in Theorem. Irreducibility conjecture. The polynomial $\mathcal{R}_F$ is…

Let $p$ be a polynomial hyperbolic with respect to $e\in \mathbb{R}^n$, and let $\Lambda_+(p,e)$ denote its hyperbolicity cone. A polynomial has a definite determinantal representa…

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

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

Let $p$ be a polynomial on $\mathbb{R}^3$ that is hyperbolic of degree $d$ with respect to $e=(1,0,0)$ and satisfies $p(e)=1$. Let $I$ denote the $d\times d$ identity matrix and le…

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…

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…

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…

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…

A homogeneous polynomial $h\in\mathbb{R}[x_1,\ldots,x_n]$ is hyperbolic with respect to $e\in\mathbb{R}^n$ if $h(e)>0$ and, for every $a\in\mathbb{R}^n$, the polynomial $h(te-a)$ h…

Let $q\geq 1$, and let $st enmathbb{R}$ and $vtinmathbb{R}^m$ for $tin [w]$ satisfy … for every $x\in\mathbb{R}^m$. Positivity-induced triangle inequality. For all…

Let $p\in \mathbb{R}[x_1,\dots,x_n]$ be a complete hyperbolic polynomial with hyperbolicity cone $C$. For a real $\alpha>n$, let $q_{\alpha}$ be the function defined by … where…

Let $p\in\mathbb{N}^*$ and let $P,Q\in\mathbb{R}[X]$ have degree $p$. Derivative-convolution conjecture. If $P$ and $Q$ are real-rooted, then … is real-rooted as well. This is pres…

For $n,p\in\mathbb{N}$, define the $p$-homogeneous semi-symmetric polynomial … for $x\in\mathbb{R}^n$ and $\lambda\in\mathbb{R}^p$. Let $\mathbb{I}$ denote the all-ones direction,…