10 problems

Let $n\in\mathbb{N}-\{0\}$ and let $K$ be a field. Let $S$ be an affine subspace of $M(n\times n,K)$ such that every element of $S$ is nilpotent, and assume that each element of…

Let $Q$ be a stable partition of length $\ell$, let $J_Q$ be a Jordan matrix of type $Q$, and let $\mathcal N_{J_Q}$ be the variety of nilpotent matrices commuting with $J_Q$. For…

For integers $g\ge 2$ and $n\ge 1$, let $A_g(n,q)$ be the polynomial associated in the paper with the counting of tuples of nilpotent matrices under simultaneous conjugation. Write…

Let $F:K^d\rightarrow K^d$ be a polynomial map with $F_i=X_i+H_i$, where … and … for $1\leq i\leq d$. Write $H=(H_1,\dots,H_d)$ and let $JH$ be its Jacobian matrix. Define polynomi…

Let $P$ be a partition, let $\mathcal D_P$ be the associated poset, and let $C_a$ be a maximum-length U-chain in $\mathcal D_P$. Removing $C_a$ gives a partition $P'(P,a)$; recursi…

Fix $Q=(u,u-r)$ with $u>r\ge2$, put $B=J_Q$, and let $E(\mu,\nu)$ and $X_s(k_1,k_2)$ be the coordinate expressions defined in the source. All-table loci equations conjecture. For…

Fix $Q=(u,u-r)$ with $u>r\ge2$, put $B=J_Q$, and use coordinates $a_1,\ldots,a_{u-1},b_1,\ldots,b_{u-r-1}$ on $\mathcal U_B$. For $\mu\in\{0,1,\ldots,u-1\}$ and…

Fix $Q=(u,u-r)$ with $u>r\ge 2$, put $B=J_Q$, and let $\mathcal U_B$ be the maximal nilpotent subalgebra of the centralizer of $B$. For a partition $P$, let…

Nilpotence conjecture. One has

Let $M$ be a square matrix of size $m$ whose entries are functions from a set $S$ to a reduced noncommutative ring $R$. For $v\in S$, write $M|_v$ for the matrix obtained by evalua…