18 problems

Generation conjecture. The algebra of all invariant operators for the $sp(2n+2)$ action on $R_{\delta}$ is generated by the operators $X$, $i(\alpha)$ and the projectors onto the s…

Unique maximality conjecture. There is a unique maximal ${\cal X}$-matroid on $K_n$.

Border Comon's conjecture for minimal border rank. If

Let $\alpha=(0<\alpha_0\leq\cdots\leq\alpha_{d-1})\in\mathbbm{N}^{d+1}$. For the monomial $x_0^{\alpha_0}\cdots x_{d-1}^{\alpha_{d-1}}$, its symmetric border rank is the least numb…

Finite-fiber conjecture for symmetric tensors. All fibers of $\Phi_{\mathrm{sym}}$ are finite.

Definiteness conjecture. Then $\mathcal{A}$ is neither a positive semidefinite tensor nor a negative semidefinite tensor.

Let $I$ be an index set, let $d$ be a positive integer, let $\mathcal{C}\in(\mathbb{R}^{I})^{\otimes d}$ be a symmetric tensor, and let…

Let $H=(V,E)$ be a $3$-uniform hypergraph with $n$ vertices. Quadratic bound conjecture. Its pair-matching decomposition number satisfies … This bound would follow from the precedi…

Let $n\geq 2$. For a homogeneous polynomial $f\in\mathbb{C}[x_0,\dots,x_n]_d$, let $E(f)$ denote its eigenscheme. Consider a set…

Let $x_0,\ldots,x_n$ be variables and let $0\leq\alpha_0\leq\cdots\leq\alpha_n$. Write $\operatorname{bsrk}$ for border symmetric rank. Border symmetric rank conjecture. For these…

Let a binary form of degree $kd$ be expressed as a sum of $k$-th powers of binary forms of degree $d$, and let $\mathrm{rk}^{\max}_k(2,kd)$ denote the maximum of the resulting $k$-…

Let $\mathcal{H}\in\operatorname{H}^m(\mathbb{C}^n)$ be any Hankel tensor, where $\operatorname{rank}(\mathcal{H})$ denotes its tensor rank and $\operatorname{rank}_S(\mathcal{H})$…

Let $T\in\mathbb{C}^{I\times I\times I}$ be a symmetric tensor, and let $\mathcal{M}\subset\mathbb{C}^{I\times I}$ be a set of symmetric rank-one matrices. Let $\mathcal{T}$ be obt…

Fix $n\in\mathord{\mathbb N}$ with $2<n$, and let ${\cal S}\in\mathrm{S}^d\mathord{\mathbb C}^n$ be nonsingular. Let $\mathbf H$ denote the polynomial map whose fixed points are us…

Let $S^d(\mathbb C^n)$ be the space of symmetric tensors, and let the odeco variety be the Zariski closure of the orthogonally decomposable tensors. For a tensor…

Generic uniqueness conjecture. Every $\mathcal{T}\in\mathbb{R}^{n_1\times\cdots\times n_d}$ has a unique rank-one approximation, unless $\mathcal{T}$ lies on a subvariety; moreover…

Let $V$ be a vector space and let $p\in\mathbb P S^dV\vartriangleleft\mathbb P(V^{\otimes d})$. Let $R_{v_d(\mathbb P V)}(p)$ be the symmetric rank of $p$, and let…

Let $\mathcal{Y}_{r}$ be the set of symmetric tensors of symmetric rank at most $r$, and let $\overline{\mathcal{Y}}_{r}$ denote its closure. Let $R_{\mathsf{S}}$ be the maximum sy…