On the b-function with respect to weights of annihilating ideals in the Weyl algebra

TL;DR

This paper investigates the properties of the b-function associated with weights of the annihilating ideal of a polynomial raised to a power, providing explicit expressions in certain cases.

Contribution

It introduces new insights into the b-function related to weighted annihilating ideals in the Weyl algebra, with explicit formulas in special cases.

Findings

Explicit expressions for the b-function in particular cases

Analysis of properties of the b-function with respect to weights

Enhanced understanding of annihilating ideals in the Weyl algebra

Abstract

Given a polynomial $f\in\mathbb{C}[x_1,\ldots,x_n]$ and an integer $\ell\in\mathbb{Z}$, we study some properties of the b-function with respect to weights of the annihilating ideal Ann$(f^\ell)$. In some particular cases the expression of the b-function is given explicitly.