26 problems

Beasley–Laffey's maximal-rank conjecture. The maximal possible rank of an $n\times n$ Euclidean distance matrix is $n$.

Let $X$ and $Y$ be finite sets, and let $f:X\times Y\rightarrow \{-1,1\}$ be a Boolean function whose communication matrix has rank $r$. Write $\operatorname{CC}^{\det}(f)$ for its…

Let $m=ngeq4$, and let $\mathcal{S}_n$ denote the real symmetric $n\times n$ matrices. A linear operator $\mathcal{L}:\mathcal{S}_n\to\mathcal{S}_n$ preserves cp-rank if it maps ev…

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,…

Let $M$ be a Boolean matrix. Let ${\mathrm{p}}(M)$ be its partitioning number, and let $\operatorname{srr}(M)$ be the minimum number of primitive matrices needed to express $M$ as…

Let $M$ be a Boolean matrix. Write ${\mathrm{r}}(M)$ for its rank over the reals, ${\mathrm{p}}(M)$ for its partitioning number, and let the communication complexity of $M$ be the…

Let $A$ be a binary matrix, and let $\operatorname{D}(A)$ denote its deterministic communication complexity and $\operatorname{rank}(A)$ its rank. Log-rank conjecture. There is an…

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

Let $A$ and $B$ be tuples of $m$ matrices, and let $f$ range over all noncommutative polynomials in $m$ variables. Write $\operatorname{rk} f(A)$ for the rank of the matrix obtaine…

Cadney–Linden–Winter rank conjecture. The inequality

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…

Hankel rank conjecture. Then

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…

Log-rank conjecture. There is a polylogarithmic upper bound

Let $A$ be a binary matrix, let $D(A)$ denote its deterministic communication complexity, and let $R_{\mathbb{R}}(A)$ denote its real rank. Log-rank conjecture. … If true, this wou…

Let $M$ be a $0/1$-matrix, and let $\mathrm{rk}(M)$ denote its rank. A deterministic communication protocol computes $M$ when it determines every matrix entry, and the nonnegative…

Let $G$ be a non-bipartite graph, and let $\mathcal{P}(\mathcal{S}_{r+}^n)$ denote its rank-restricted coordinate shadow at rank bound $r$. Tripartite closure conjecture. Tripartit…

Let $N(r,L)$ denote the maximum dimension of an $L$-matrix of rank at most $r$, and let $L$ be a set of permitted entries. Quadratic growth conjecture. The exponent $3/2$ in Theore…

Let $L=\{\alpha_1,\dotsc,\alpha_k\}$ be a set, and call $(A_1,\dotsc,A_k)\in\mathbb{Z}^k$ a primitive linear relation on $L$ if … Let $N(r,L)$ denote the extremal function from the…

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

Log Rank Conjecture. The deterministic communication complexity of $F$ is upper bounded by