Equivariant Poincar\'e series and topology of valuations

TL;DR

This paper demonstrates that the equivariant Poincaré series, a power series incorporating group actions, can determine the topology of a collection of valuations, enhancing understanding of their structure in algebraic geometry.

Contribution

It establishes that, except for simple cases, the equivariant Poincaré series fully encodes the equivariant topology of valuations.

Findings

Equivariant Poincaré series determines valuation topology.

The series encodes group action information.

Results hold with simple exceptions.

Abstract

The equivariant with respect to a finite group action Poincar\'e series of a collection of $r$ valuations was defined earlier as a power series in $r$ variables with the coefficients from a modification of the Burnside ring of the group. Here we show that (modulo simple exceptions) the equivariant Poincar\'e series determines the equivariant topology of the collection of valuations.