12 problems

Rank-deficient permutation coverage conjecture. For $V$ generic Toeplitz, every permutation $P$ for which

Let $M$ be a Boolean matrix, and let a monochromatic rectangle be a submatrix all of whose entries have the same value. Let ${\mathrm{r}}(M)$ denote the rank of $M$ over the reals,…

Rank Nullstellensatz conjecture. The following are equivalent: (i) there is $K\in\mathbb{N}$ such that for every $n\in\mathbb{N}$ and every…

Makar-Limanov's conjecture.

D--Kakde's Matrix Coefficient Conjecture. If $\det(M)=0$, then there exist nonzero vectors $w,v\in\mathbf Q^n$ such that

$p$-adic Structural Rank Conjecture. For every

Structural Rank Conjecture. For every $M\in M_{m\times n}(\mathscr L+\mathbf Q)$, its rank equals its structural rank.

Parallel $k$-partition conjecture. For every fixed integer $k>1$,

Let $t$ be an integer satisfying $t\equiv 0,1,2\pmod{6}$, and let $r$ be an integer with $r>\lfloor t/6\rfloor$. The extended cube graphs are denoted by…

Let $\mathbb{F}$ be a field of zero characteristics. For positive integers $k,n$, let $M_{k,n}(\mathbb{F})$ be the set of $k$-by-$n$ matrices over $\mathbb{F}$, and let…

Let $f:X\times Y\to\{0,1\}$ be a Boolean function, and let $\mathrm{CC}^{\mathrm{det}}(f)$ denote its deterministic communication complexity. Let $M_f$ be its communication matrix…

Li et al.'s conjecture. If $r\geqslant t-2$, then the minimum rank of $S$ over the rationals is also $r$.