10 problems

Let $R$ be a commutative ring and let $P$ be a Col-divisible polytope of arbitrary dimension. Write $K_i(R,P)$ for its associated $K$-groups, and let ${\mathfrak c}(P)\leq\dim P$ b…

Let $R$ be a commutative ring with identity. For a positive integer $s$, let $p_{2s}(R)$ denote the least number of $2s$-th powers needed in the relevant representation defining th…

Weighted Erdős–Burgess lower-bound conjecture.

Let $A$ be a commutative ring with unit in which $2\neq0$, let $(N,Q)$ be a quadratic module over $A$, and set $\mathcal{C}:=\mathcal{C}(N,Q)$. Let $e$ be the idempotent used to fo…

Let $R$ be a commutative ring and let $(M,q)$ be a quadratic module over $R$. Write $M^{\perp}$ for the radical of the polar form of $q$, and let an orthogonal map mean an $R$-line…

Variable-exponent central power-difference conjecture. Under these conditions, $R$ is commutative.

Let $R$ be a commutative unitary ring. Write $J(R)$ for its Jacobson radical, and let $\textsc{I}(\mathcal{S}_R)$ denote the Erdős–Burgess constant of the multiplicative semigroup…

Let $\mathbf U$ be the quantum group and let $\mathbb P=P\otimes_R\mathcal F$ be as above, where $\mathcal F$ is a finite-dimensional $\mathbf U$-module. Let…

Nilpotence conjecture. One has

Beck's conjecture. For every commutative ring $R$ with unity,