10 problems

Let $V$ be a finite-dimensional vector space, let $f\in S^3V$, and let $E\subseteq V^*$ be a linear subspace. Write $f|_E$ for the restriction of $f$ to $E$, and let…

Let $d\geq 2$, let $\mathbb{K}$ be a field, let $\mathsf{W}\subseteq\mathbb{K}^{n_1}\otimes\cdots\otimes\mathbb{K}^{n_d}$ be a linear subspace, and let $k$ be a positive integer sa…

Let $d\geq 2$, let $\mathbb{K}$ be a field, and let $\mathsf{W}\subseteq\mathbb{K}^{n_1}\otimes\cdots\otimes\mathbb{K}^{n_d}$ be a linear subspace. Write $\overline{\mathbb{K}}$ fo…

Bounded-dimensional support conjecture. There exists a positive integer $C(d,k)$ such that, for every order-$d$ tensor $T$ over an arbitrary field $\mathbb{F}$ with slice rank $k$,…

Let $G$ be a finite group, let $M(G)$ denote its matching number, and define the minimum slice rank … where the minimum is over algebraically closed fields $k$ and $T_{k[G]}$ is th…

Let $d\ge2$ be an integer and let $R$ be a non-empty family of partitions on $\lbrack d\rbrack$. Let $\operatorname{\mathit{R}rk}$ denote the rank associated with $R$. Linear bound…

For $d\geq 3$, let … be the Shioda polynomial in $n+2$ variables. For even $n$, write $\operatorname{sr}(S_{n,d})$ for its slice rank and $\operatorname{B}_{\mathrm{hom}}(S_{n,d})$…

Conjecture B. There exists a function $c(r,d)$ such that

Strength–slice-rank conjecture. The strength and the slice rank of a general form in $\mathcal{S}_d$ are equal.

Let $f$ be a general form of degree $d\geq2$ in $n+1$ variables. Write $\mathrm{str}(f)$ for its strength and $\mathrm{sl.rk}_{n,d}^\circ$ for the general slice rank in degree $d$…