# Bernstein-Sato functional equations, $V$-filtrations, and multiplier ideals of direct summands

## TL;DR

This paper extends the theory of Bernstein-Sato polynomials, V-filtrations, and multiplier ideals to nonregular rings and direct summands, providing new tools for studying singular algebraic varieties.

## Contribution

It constructs D-modules for nonregular rings, proves the existence of Bernstein-Sato polynomials for direct summands, and develops V-filtrations in this broader context.

## Key findings

- Bernstein-Sato polynomial exists for direct summands of polynomial rings.
- V-filtrations are established for nonregular rings including toric and determinantal rings.
- Relations among multiplier and Hodge ideals are extended to singular settings.

## Abstract

This paper investigates the existence and properties of a Bernstein-Sato functional equation in nonregular settings. In particular, we construct $D$-modules in which such formal equations can be studied. The existence of the Bernstein-Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of $V$-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings.