28 problems

Comon's conjecture. The Waring rank of a symmetric tensor equals its tensor rank, and the border Waring rank equals the border tensor rank.

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

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…

Qi's conjecture. The bound

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.

Let $\mu^{(k)}$ be a monotonically decreasing sequence with $\lim_{k\rightarrow\infty}\mu^{(k)}=0$, and let $({\mathbf{X}}^{(k)},{\mathbf{S}}^{(k)})$ be points on the central path…

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…

Positive-matching conjecture. The set $E_{l_1,l_2}$ is a positive matching.

Condition-number equality conjecture.

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 $T$ be a symmetric tensor of arbitrary order, let $T_{i_1,\dots,i_k}$ denote one of its entries, and let $\lambda_{maxabs}$ denote the largest absolute value of an eigenvalue o…

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 $T\in\mathbb{C}^{I\times I\times I}$ 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…

Seven-invariant conjecture. If $\mathcal{A}\in T(\mathbb{H})$ is symmetric, then it has seven independent invariants.

Let ${\cal S}\in\mathrm{S}^d\mathord{\mathbb C}^n$ be a symmetric tensor. Nonsingularity-decision conjecture. With probability $1$, one can decide in time polynomial in $d$ whether…

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 $n\geq 3$ be fixed, and let ${\cal S}\in\mathrm{S}^d\mathord{\mathbb C}^n$ be a symmetric $d$-qudit. For most such tensors, consider the spectral norm $\|{\cal S}\|_{\sigma}$.…

Comon–Ottaviani conjecture. For every $d$, all ranks

Let $n-1,d-1\in\mathord{\mathbb N}$, let $k\in[d-1]$, and let ${\cal S}\in\mathrm{S}^d\mathord{\mathbb F}^n$. A best $k$-approximation of a tensor is an optimal orthogonal projecti…

Let $d\ge 3$, let $\mathord{\mathbb F}=\mathord{\mathbb R}$ or $\mathord{\mathbb C}$, and let ${\cal S}\in\mathrm{S}^d\mathord{\mathbb F}^n$. Write…

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…