13 problems

Uniqueness conjecture. There exists exactly one Leibniz superalgebra of type $(n+1,n)$ with nilindex equal to $2n$.

Let $\mathfrak{O}$ be the coefficient ring, let $A$ be an $n$-dimensional nilpotent $\mathfrak{O}$-algebra, and write $\zeta_A(t)$ and $\zeta^{\lhd}_A(t)$ for its subalgebra and id…

Optimal exponent conjecture. The smallest $m\in\mathbb{N}$ such that

Let $\mathbf{A}$ be a finite nilpotent Maltsev algebra. A circuit $C_n(x_1,\ldots,x_n)$ is constant and absorbing when it is constant as a function and has the absorbing property;…

The length bound conjecture. Let $\mathcal{N}$ be a $k$-dimensional $(A,C)$-commutative $n$-ary algebra. Then its length is at most

Dubnov–Ivanov–Nagata–Higman conjecture.

Degree and limiting-value conjecture. If $({\mathcal{L}},{\mathcal{E}})$ is almost uniform, so that…

Let $n\in\mathbb{N}$, let $k\geqslant 2$, and let $\varphi_k$ denote the constant used in the paper's asymptotic bound. A $k$-step nilpotent algebra is a quadratic $\mathbb{K}$-alg…

Let $\cA$ be a nilpotent $\fo$-algebra that is free of rank $d$ as an $\fo$-module, where $\fo$ is the ring of integers in a number field $k$. Let $K\supset k$ be a $p$-adic field…

Let $\cA$ be a Lie algebra or a non-unital associative $\fo$-algebra, free of rank $d$ as an $\fo$-module, where $\fo$ is the ring of integers in a number field $k$, and suppose th…

Let $S$ be a finite nilpotent commutative semigroup with zero element $0$. For a positive integer $n$, write $S^{(n)}$ for the set of $n$-th powers of elements of $S$. The semigrou…

Let $F$ be a perfect field of characteristic $p>0$, let $R$ be a finite-dimensional nilpotent commutative algebra over $F$, and let $R^{(p)}$ be the image of the $p$-th power map o…

The conjectured lower-bound equality. One has