Computing topological zeta functions of groups, algebras, and modules, I

TL;DR

This paper introduces new convex-geometric methods to compute topological zeta functions for nilpotent groups, algebras, and modules, enabling systematic analysis of their properties.

Contribution

It develops explicit formulas for p-adic integrals and applies them to compute and analyze topological zeta functions of various algebraic structures.

Findings

First systematic computation of topological zeta functions for algebras

Explicit convex-geometric formulas for p-adic integrals

Enhanced understanding of properties of zeta functions

Abstract

We develop techniques for computing zeta functions associated with nilpotent groups, not necessarily associative algebras, and modules, as well as Igusa-type zeta functions. At the heart of our method lies an explicit convex-geometric formula for a class of $p$-adic integrals under non-degeneracy conditions with respect to associated Newton polytopes. Our techniques prove to be especially useful for the computation of topological zeta functions associated with algebras, resulting in the first systematic investigation of their properties.