Ideal growth in nilpotent rings and zeta functions of quiver representations

In a seminal paper, Grunewald, Segal and Smith (1988) introduced zeta functions of groups and rings, enumerating various types of sub-objects, such as subgroups or two-sided ideals of finite index. Computations of these functions involve other enumeration problems in algebra and combinatorics.

Our main focus will be on zeta functions enumerating ideals of finite (additive) index, in nilpotent rings of class 2. It is well known that in this case there is a decomposition into an Euler product of local factors indexed by primes. The local factors are rational functions over the rationals, but their explicit computation is usually a very hard problem.

We show that the complexity of computing the ideal zeta function of an amalgamated direct power of such rings does not increase by the amalgamation. More generally, we prove this for the zeta functions of quiver representations introduced by Lee and Voll (2021). The proof of rationality of the local factors relies on techniques from model theory, and their explicit computation uses tools from algebraic combinatorics. Such methods are likely to be applicable to other combinatorial enumeration problems in which one wants to prove polynomiality, rationality or uniformity.

This is a joint work with Michael Schein.
No prior experience with zeta functions of groups and rings will be assumed.