Enumerating graded ideals in graded rings associated to free nilpotent Lie rings

Abstract

We compute the zeta functions enumerating graded ideals in the graded Lie rings associated with the free $d$-generator Lie rings ${\mathfrak{f}}_{c,d}$ of nilpotency class $c$ for all $c\le 2$ and for $(c,d)\in \{(3,3),(3,2),(4,2)\}$. We apply our computations to obtain information about $\mathfrak{p}$-adic, reduced, and topological zeta functions, in particular pertaining to their degrees and some special values.

28 pages