6 problems

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

Border Comon's conjecture for minimal border rank. If

Let $m=x_0^{a_0}\cdots x_n^{a_n}$ be a monomial. Write $d=\sum_{i=0}^n a_i$, and let $\underline{\mathrm{cR}}_{d}(m)$ and $\underline{\mathrm{R}}_{d}(m)$ denote the border cactus r…

Even-degree border-rank lower-bound conjecture. With the same Koszul Young flattening method, one can show, for some constant $C$ and for $n$ sufficiently large,

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 $M_{\langle n,n,n\rangle}$ denote the tensor encoding multiplication of two $n\times n$ matrices, and let $\underline{R}(M_{\langle n,n,n\rangle})$ be its border rank. For ever…