Poincar\'e series of multiplier and test ideals

TL;DR

This paper proves the rationality of Poincaré series for multiplier and test ideals across all dimensions, extending previous surface-specific results through a unified Hilbert function framework.

Contribution

It introduces a new theory of real-indexed Hilbert functions to establish the rationality of Poincaré series for multiplier and test ideals in any dimension.

Findings

Proves rationality of Poincaré series in all dimensions.

Extends results from surfaces to higher dimensions.

Develops a unified Hilbert function approach.

Abstract

We prove the rationality of the Poincar\'e series of multiplier ideals in any dimension and thus extending the main results for surfaces of Galindo and Monserrat and Alberich-Carrami\~nana et al. Our results also hold for Poincar\'e series of test ideals. In order to do so, we introduce a theory of Hilbert functions indexed over $\mathbb{R}$ which gives an unified treatment of both cases.