On the b-function with respect to weights of annihilating ideals in the Weyl algebra
Helena Cobo
TL;DR
This work studies the b-function of the annihilating ideals Ann(f^ℓ) in the Weyl algebra with respect to weights. It builds a framework based on Gröbner deformations via the initial ideals in_{(-ω,ω)} and the s-parametric annihilator Ann(f^s), enabling algebraic control over b_{Ann(f^ℓ),ω}(s). Key results include that for any nonzero ω the quantity ℓ ord_f(ω) is a root of b_{Ann(f^ℓ),ω}(s), with b_{Ann(f^ℓ),ω}(s) = s − ℓ ord_f(ω) when ℓ ∈ N, and explicit formulas in the homogeneous two-variable case and for curves f = x^p + y^q when ℓ ≤ 0. The paper also analyzes deformations, showing that the weighted b-function is not a topological invariant in general, though it remains constant under certain quasi-homogeneous and μ-constant deformations in specific settings. These results illuminate geometric structure behind weighted b-functions and their relevance to Bernstein–Sato theory and localization problems.
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.