46 problems

Let $\omega$ denote the exponent of matrix multiplication, equivalently the asymptotic exponent governing the border rank of the matrix multiplication tensors. Exponent-two conject…

Let $\mathcal{P}_1\in U_1\otimes\cdots\otimes U_n$ and $\mathcal{P}_2\in V_1\otimes\cdots\otimes V_n$ be persistent tensors, where $U_1,\ldots,U_n$ and $V_1,\ldots,V_n$ are vector…

Let $\mathbb{k}$ be the underlying field, and let $T\in\mathbb{k}^a\otimes\mathbb{k}^b\otimes\mathbb{k}^c$ be a tensor that is tight and concise. Here, concise means that the flatt…

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 $A_1,B_1,C_1,A_2,B_2,C_2$ be finite-dimensional vector spaces over a field $\mathbb{K}$, and let $T_1\in A_1\otimes B_1\otimes C_1$ and $T_2\in A_2\otimes B_2\otimes C_2$. Thei…

Let $d\geq 2$, let $q$ be a prime power, and let $T\in\mathbb{F}_q^{n_1}\otimes\cdots\otimes\mathbb{F}_q^{n_d}$, with $\operatorname{PR}_{\mathbb{F}_q}(T)$ and…

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 $S$ be the polynomial ring in $n$ variables over the complex numbers. For positive integers $k,d,n$, let $\operatorname{rk}_k^\circ(kd,n)$ denote the $k$-rank of a general form…

Let $m\geq 5$ be odd, and let $\mathcal{H}\in\operatorname{H}^m(\mathbb{C}^n)$ be a generic Hankel tensor. The generic Hankel rank conjecture. The answer to whether … is yes. The e…

Let $f,g\in\mathbb{C}[x_1,\ldots,x_n]$ be homogeneous polynomials of degrees $d$ and $d-1$, respectively. Let $u$ be a new variable, and let $\ell[x_1,\ldots,x_n]$ be the set of al…

Let $\mathbb O$ be the octonion algebra, viewed as an $8$-dimensional real algebra, and let $T_{\mathbb O}\in\mathbb R^8\otimes\mathbb R^8\otimes\mathbb R^8$ be its multiplication…

Two-orthogonal uniqueness and stabilization conjecture. A generic two-orthogonal tensor in $\mathbb{R}^{n_1}\otimes\cdots\otimes\mathbb{R}^{n_d}$ has a unique two-orthogonal decomp…

Border Comon's conjecture for minimal border rank. If

Let $d\geq 2$, let $q$ be a prime power, and let $T\in\mathbb{F}_q^{n_1}\otimes\cdots\otimes\mathbb{F}_q^{n_d}$. For $n\geq 1$, let $T^{\oplus n}$ denote the $n$-fold direct sum of…

Let $d\geq 2$, let $q$ be a prime power, and let $T\in\mathbb{F}_q^{n_1}\otimes\cdots\otimes\mathbb{F}_q^{n_d}$. Write $\overline{\mathbb{F}}_q$ for an algebraic closure of…

Let $\mathcal{H}=\mathbb{C}^{d_1}\otimes\cdots\otimes\mathbb{C}^{d_n}$ be the Hilbert space of multipartite quantum states. The expected tensor rank is the smallest $k$ for which…

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

Let $\mathbb{F}$ be the underlying field, and consider concise tensors in … Write $\sigma(d)$ for the worst-case exponent of this space. Extended asymptotic rank conjecture. For al…

Generic simultaneous Waring rank conjecture. The complex simultaneous Waring rank of a general polynomial vector is

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…

Rank-separation conjecture. There exist no functions $f,g,h\colon\mathbb{N}\to\mathbb{N}$ such that, for every $p\in\mathbb{R}[x,y]$, independently of the degree of $p$,

Let $\mathbf k$ be a field, let $\bar{\mathbf k}$ be its algebraic closure, let $V_1,\ldots,V_d$ be $\mathbf k$-vector spaces, and let $L\subset (V_1\otimes\cdots\otimes V_d)^\vee$…

Let $d\geq 2$, let $\mathbb F_q$ be a finite field, and let $P$ be a multilinear polynomial of degree $d$. Denote by $r_{\mathbb F_q}(P)$ its Schmidt rank and by…

Let $\mathbf k$ be a field, let $\bar{\mathbf k}$ be its algebraic closure, and let $P$ be a multilinear $\mathbf k$-polynomial of degree $d\geq 2$. Write $r_{\mathbf k}(P)$ for it…

Non-viability criterion. If