35 problems

Estes–Guralnick conjecture. Every such polynomial $f$ is the minimal polynomial of some integer symmetric matrix.

Luschny's permanent conjecture. Luschny conjectured that

Let $F_n$ be the $n$th Fibonacci number, let $e(n,i,k)$ be the exponent used to determine the sign in the source, and let $d_k(r)$ be the Fibonomial coefficient corresponding to…

For a positive integer $r$, let $b_k(r)=\binom{k+r-1}{r}$, and let $R_n(b_k(r))$ denote the corresponding reciprocal Hankel matrix. Integrality criterion conjecture. The inverse of…

Let $F_n$ denote the $n$th Fibonacci number. A Fibonomial coefficient is obtained by replacing the binomial coefficients in the corresponding reciprocal Hankel matrices with Fibono…

Bounded-correction conjecture. For every such 4-tuple,

Villarreal's determinant conjecture. If $\alpha_0(\mathcal C')=\beta_1(\mathcal C')$ for all minors $\mathcal C'$ of $\mathcal C$ and $x^{v_1},\dots,x^{v_q}$ have degree $d\geq 2$,…

Asymptotic counting conjecture. The lower bound gives the true order of growth:

Let $n$ be a positive integer and let $B$ tend to infinity. Choose a matrix from the set of matrices in $\operatorname{SL}(n,\mathbb{Z})$ whose coefficients have absolute value at…

Let $r\geq 2$ and let $A\in \operatorname{Mat}_r(\mathbf Z)$ be nonsingular. Define $\gcd(A-I)$ as the greatest common divisor of the entries of $A-I$, and call $A$ primitive when…

Let $\lambda$ be a partition with $n$ parts, and let $M=(m_{i,j})_{1\le i,j \le n}\in\mathcal{M}_\lambda$ be a minimal matrix of shape $\lambda$. A matrix is Hankel when its entrie…

Let $G$ be the topological ordering of a graph on $[n]$, and let $B$ be the matrix over ${\mathbb{Z}}/p{\mathbb{Z}}$ given by … Let $r_p$ be the rank of $B$ over…

Zhu–Chen–Hu conjecture. When $a=b$, the three factorizations $(S^a)\mid(S^b)$, $(S^a)\mid[S^b]$, and $[S^a]\mid[S^b]$ in the corresponding integer matrix ring hold if and only if…

Selberg error-term conjecture. For every $\varepsilon>0$,

The column number functions $\operatorname{g}(\Delta,r)$, $\operatorname{s}(\Delta,r)$, and $\operatorname{h}(\Delta,r)$ exhibit erratic behavior when one parameter is fixed and th…

For a polynomial $f\in\mathbb{Z}[X]$, let $P_n(H;f)$ denote the number of matrices $A\in\mathcal{M}_n(\mathbb{Z};H)$ with characteristic polynomial $f$. Characteristic-polynomial s…

Eventual formula conjecture. There is a constant $\delta_2\in\mathbb{Z}_{>0}$ such that, for every $\Delta\geq\delta_2$,

Monotonicity conjecture. The function

Bounded-excess conjecture. There is a constant $c>0$ such that

Maximal-column conjecture. The identity

Let $B$ be an $m\times n$ matrix with integer entries. Write $D_k(B)$ for its $k$th determinantal divisor and $D_k^*(B)$ for the corresponding complementary determinantal divisor.…

Let $R_n(H;f)$ denote the number of matrices $A\in\mathcal M_n(\mathbb Z;H)$ with characteristic polynomial $f\in\mathbb Z[X]$, where $\mathcal M_n(\mathbb Z;H)$ is the set of…

Lee et al.'s conjecture. For every $cDelta,m cin cmathbb{Z}_{>0}$,

Let $M$ be a $p\times q$ integer matrix with $p\geq 2$. For $n\in\mathbb{N}$, let $m(M,n)$ be the minimum cardinality of a subset $S\subseteq [q]^n$ such that the associated map…

Nilpotence conjecture. For all $a\geqslant 0$, $b\geqslant 0$, and $k\geqslant 0$, there exists an integer $N\geqslant 0$ such that every entry of