10 problems

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…

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 $f\in\mathbb{R}[x_1,\ldots,x_m]$ 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 $d$.

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

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 $p\in\mathbb{R}[x_1,\ldots,x_n]$ 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…