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

TL;DR

This paper computes zeta functions counting graded ideals in specific graded Lie rings related to free nilpotent Lie rings, providing detailed information on their properties for certain classes and generator counts.

Contribution

It offers explicit calculations of zeta functions for graded ideals in free nilpotent Lie rings of specified classes and generator counts, extending previous understanding.

Findings

Computed zeta functions for c ≤ 2 and specific (c,d) cases

Analyzed degrees and special values of p-adic, reduced, and topological zeta functions

Provided new explicit formulas and insights into the structure of graded ideals

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 $c\leq2$ 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.