9 problems

Let $c\in\mathbb{N}$ and $d\in\mathbb{N}_{\geq2}$, let $W=W_d$ be the Witt function, and let $\mathfrak{o}$ be a finite extension of $\mathbb{Z}_p$. Write…

Let $c\in\mathbb{N}$ and $d\in\mathbb{N}_{\geq2}$, let $W=W_d$ be the Witt function, and let $\mathfrak{f}_{c,d}$ be the free nilpotent Lie ring of class $c$ on $d$ generators. Wri…

Assume the uniformity conjecture, so that the reduced graded ideal zeta function $\zeta_{\mathfrak{f}_{c,d},\operatorname{red}}^{\triangleleft_{\operatorname{gr}}}(Y)$ is defined.…

Let $c\in\mathbb{N}$ and $d\in\mathbb{N}_{\geq2}$, let $\mathfrak{f}_{c,d}$ be the free nilpotent Lie ring of class $c$ on $d$ generators, and put…

Assume the uniformity conjecture, and define the reduced graded ideal zeta function by … Let $r=\sum_{i=1}^cW_d(i)$ and $[N]_0=\{0,\ldots,N\}$. Reduced zeta-function conjecture. Th…

Let $c\in\mathbb{N}$ and $d\in\mathbb{N}_{\geq2}$, let $W=W_d$ be the Witt function, and put $r=\operatorname{rk}_{\mathbb{Z}}(\mathfrak{f}_{c,d})=\sum_{i=1}^cW(i)$. For a finite e…

Let $c\in\mathbb{N}$ and $d\in\mathbb{N}_{\geq 2}$. Let $\mathfrak{f}_{c,d}$ be the free nilpotent Lie ring of class $c$ on $d$ generators, and let $\mathfrak{o}$ be a finite exten…

Let $L$ be a class-$2$-nilpotent Lie ring with $\operatorname{rk}(L/L')=d$, $\operatorname{rk}(Z(L))=m$, and $\operatorname{rk}(L/Z(L))=r$. Let $\alpha^{\triangleleft}$ be the absc…

Let $L$ be a class-$c$-nilpotent Lie ring of rank $n$, with upper central series $(Z_i(L))_{i=0}^c$. Set $n_i:=\operatorname{rk}(L/Z_i(L))$, so that $n_0=n=\operatorname{rk}(L)$. L…