71 problems

Triple-coincidence conjecture. For every integer $n\geq4$, no three distinct pieces in $\mathcal{P}'$ share the same strength on the $n\times n$ board.

For every finite associative unital ring $R$, let $N(R):=\{f(x)\in R[x]\mid f(a)=0\text{ for every }a\in R\}$ be the null ideal of $R$. The conjecture asserts that $N(R)$ is a two-…

For every fixed integer $m\ge 1$ and every collection of pairwise distinct nonconstant polynomials $f_1,\ldots,f_m\in\mathbb{C}[x]$, there exists an integer $d_0$ such that, for ev…

A polynomial solution $(\mathcal{A},\mathcal{B},\mathcal{C})$ is constructed for each standard Pythagorean triple, where $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are polynom…

Amitsur's conjecture. The limit exists and belongs to $\mathbb Z_+$.

Let $n\geq 2$ be an integer, let $K$ be a field, and let $K\langle X\rangle$ be the free associative algebra over $K$ on a countable set of noncommutative variables. A multilinear…

Let $m\in\mathbb{N}$ and let $x$ be a positive integer. The coefficients ${\mathbf{A}}_{m,0}, {\mathbf{A}}_{m,1},\ldots,{\mathbf{A}}_{m,m}$ are constants independent of $x$ and $k$…

Let $F$ be a field of characteristic $0$, let $H$ be a Hopf algebra over $F$, and let $A$ be a finite-dimensional associative $H$-module algebra, with codimensions … where $P_n^H$…

Let $A$ be a $W$-algebra over a field of characteristic zero. A class of algebras has the Specht property if every $T$-ideal of the class is finitely generated. The Specht conjectu…

Let $T_m(K)$ denote the algebra of upper triangular $m\times m$ matrices over a field $K$, and let $p$ be a multilinear polynomial in noncommutative variables. Fagundes–de Mello co…

Graded Amitsur conjecture. There exists a graded PI-exponent

Bahturin–Zaicev conjecture. One has

Finite basis conjecture. The algebra $M_2(K)$ does not have a finite basis of its polynomial identities.

Let $(\lambda(1),\emptyset,\lambda(3),\lambda(4))$ be a highest-weight label in the decomposition denoted by $(\lambda(1),\emptyset,\lambda(3),\lambda(4))$, where … and … Here…

Let $K\in\mathbb{N}$, let $\lambda\vdash 4K$, and write $\lambda^{(n-3K)}$ for the partition obtained by adding a first row of length $n-3K$ to $\lambda$. Let…

Fix $K\in\mathbb{N}$. Let $W_{n,n+K}$ be the representation considered in the paper, and let $p_K(x)$ and $N_K$ be as in the preceding corollary, so that $p_K(n)=\dim(W_{n,n+K})$ f…

For each $n,K\in\mathbb{N}$, let $W_{n,n+K}$ be the representation considered in the paper, and let $\dim(W_{n,n+K})$ denote its dimension. Dimension-growth conjecture. Fix…

Let $L$ be a Lie $F$-algebra acting on a finite-dimensional algebra $A$ by derivations, so that $A$ is an $L$-algebra. Let $c_n(A)$ and $c_n^L(A)$ denote the ordinary and different…

Let $\tilde{\mathfrak M}$ be the rock-paper-scissors algebra with unit over a field $\mathbb F$, and let $p$ be a homogeneous polynomial in one variable whose sum of coefficients i…

Hartley's conjecture. If the unit group $U(FG)$ satisfies a group identity, then $FG$ satisfies a polynomial identity.

Kuzmin's conjecture. The exact value of the nilpotency class is

Let $\Lambda=(\lambda_1,\dots,\lambda_d)$ be a finite sequence with sum … Define the polynomial $F_\Lambda(x)$ by $F_\emptyset(x)=1$ and, for $1\leq d=\sharp(\Lambda)<\infty$, … Fo…

Proper containment conjecture. The ideal of polynomial identities of the path algebra is properly contained in that of its principal subalgebra:

For each positive integer $n$, let $p_n(w)$ be the polynomial counting the LEGO structures made from $n$ parallel $w\times1$ tiles. Reflection-symmetry conjecture. For every…

Nonexistence conjecture. No nontrivial $G$-grading $\Gamma$ on $M_n(K)$ satisfies the primeness property for graded central polynomials.