Computing localizations iteratively

TL;DR

This paper introduces an iterative algorithm to compute the annihilator of a function's power in the Weyl algebra, facilitating the description of localizations of polynomial rings at planar curves.

Contribution

It presents a novel iterative method using truncated annihilators to compute localizations in the Weyl algebra for planar curves.

Findings

Algorithm successfully computes annihilators for planar curves.

Method improves efficiency over previous approaches.

Applicable to algebraic analysis and D-module computations.

Abstract

Let $R=\bC[\bfx]$ be a polynomial ring with complex coefficients and $\Dx = \bC<bfx,\bfp>$ be the Weyl algebra. Describing the localization $R_f = R[f^{-1}]$ for nonzero $f\in R$ as a $\Dx$-module amounts to computing the annihilator $A = \Ann(f^a)\subset \Dx$ of the cyclic generator $f^{a}$ for a suitable negative integer $a$. We construct an iterative algorithm that uses truncated annihilators to build $A$ for planar curves.