On Poincar\'e series of filtrations

TL;DR

This survey explores the concept of Poincaré series for multi-index filtrations, introducing new computational methods, generalizations via motivic integration, and connections to zeta functions and Heegaard-Floer homologies.

Contribution

It presents an alternative definition, a novel computation method using Euler characteristic integration, and extends the notion through motivic integration and related algebraic invariants.

Findings

Introduces an alternative approach to defining Poincaré series.

Develops a method for computing Poincaré series via Euler characteristic integration.

Establishes links between Poincaré series, zeta functions, and Heegaard-Floer homologies.

Abstract

In this survey one discusses the notion of the Poincar\'e series of multi-index filtrations, an alternative approach to the definition, a method of computation of the Poincar\'e series based on the notion of integration with respect to the Euler characteristic (or rather on an infinite-dimensional version of it), generalizations of the notion of the multi-variable Poincar\'e series based on the notion of the motivic integration, and relations of the latter ones with some zeta functions over finite fields and with generating series of Heegaard-Floer homologies of algebraic links.