85 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.

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 $K$ be an infinite field, and let $v$ and $w$ be tensors, with $v\boxplus w$ denoting their vertical direct sum. Strassen's additivity conjecture. Tensor rank should satisfy ……

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…

Rhodes's conjecture. The rank of $t$ is at most $2n-1$.

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 $\mathcal{P}_1\in U_1\otimes\cdots\otimes U_n$ 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 $x_0,\ldots,x_n$ 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…

Strassen's support-functional conjecture. The asymptotic spectrum of the class of tight tensors coincides with the set of support functionals $\widehat{\zeta}^{\theta}$.

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 $f,g\in\mathbb{C}[x_1,\ldots,x_n]$ 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…

Garcia–Stillman–Sturmfels conjecture. The ideal of $\sigma_2(\operatorname{Seg}(\mathbb{P}V_1\times\cdots\times\mathbb{P}V_n))$ is generated by the $3\times 3$ minors of all flatte…

GSS conjecture. Equality holds when $r=2$:

Analytic rank–partition rank conjecture. For every $d\geq 2$, there is a constant $C_d$ such that

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…

Dimension conjecture for binary tensors. For binary tensors, equality holds:

Let $W_r$ denote the set of real tensors admitting a two-orthogonal decomposition with at most $r$ summands, and let $r_g$ be the generic rank in…

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

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

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 subrank upper-bound conjecture. The generic subrank is equal to the upper bound