17 problems

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 $p\in\mathbb{R}[\mathbf{x}]$ be a real-zero (RZ) polynomial with $p(\mathbf{0})=1$. For a polynomial $f$, write $\operatorname{rcs}(f)$ for its rigidly convex set. Generalized…

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.…

Let $d\geq 1$. A homogeneous form of degree $2d$ in three variables is nonnegative if $F(a)\geq 0$ for every $a\in\mathbb{R}^3$, and it admits a positive quadratic representation i…

Buckley–Šivic's conjecture. The Robinson polynomial does not admit a positive quadratic representation. This is the negative answer to the question of whether every nonnegative ter…

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 $p(z_1,z_2)$ be a nonzero polynomial of degree $(n_1,n_2)$, and let ${\mathbb T}^2$ denote the two-dimensional torus. A matrix $J$ is a signature matrix if $J^*=J=J^{-1}$, and…

Let $f\in\mathbb{R}[x_1,\ldots,x_m]$ be multiaffine in $x_1,\ldots,x_n$, with nonzero coefficient of $x_1\cdots x_n$. For indices $i,j$, let $\Delta_{ij}(f)$ denote the correspondi…

Continuum conjecture. There is a continuum of unitarily non-equivalent MHDRs for a bivariate polynomial of degree $d$.

Let $N^*(n,d)$ denote the minimum size of a uniform determinantal representation in the setting considered in the paper, and let $N^*(3,2)=4$ be the established preceding case. The…

Let $k$ be a positive integer, and let a general form of degree $k$ in $4$ variables be a homogeneous polynomial of degree $k$. A determinant of a $k\times k$ matrix of linear form…

Factorization characterization conjecture. The polynomial $P$ is representable if and only if, for some (equivalently any) tuple of squares $\ell\in{\mathbb F}^m$,…

Characteristic-2 nonexistence conjecture. Symmetric determinantal representations for polynomials represented by weakly-skew circuits do not always exist over fields of characteris…

Let $p \in {\mathbb R}[x_1,\ldots,x_d]$ be an $RZ_{x^0}$ polynomial of degree $m$ with $p(x^0)=1$. Here, $RZ_{x^0}$ means real-zero with respect to $x^0$, and $q^{(k)}_{x^0}$ denot…

Let $p \in {\mathbb R}[x_1,\ldots,x_d]$ be an $RZ_{x^0}$ polynomial of degree $m$ with $p(x^0)=1$. Here, $RZ_{x^0}$ means real-zero with respect to $x^0$. Generalized determinantal…

Let $p\in\mathbb{R}[x_1,\ldots,x_n]$ be a real zero polynomial if, for every $x\in\mathbb{R}^n$ and $\mu\in\mathbb{C}$, $p(\mu x)=0$ implies that $\mu$ is real. Suppose that…

Let $X \subset \mathbb{P}^r$ be a projective scheme embedded by the complete linear series of a sufficiently ample line bundle, and let $\operatorname{Sec}^k(X)$ denote the Zariski…