An Algorithm to Construct A Basis for the Module of Logarithmic Vector Fields

TL;DR

This paper presents an algorithm to construct a homogeneous basis for the module of logarithmic vector fields associated with finite collections of weighted hyperplanes, focusing on two-dimensional cases.

Contribution

The paper introduces a new algorithm for explicitly constructing a basis for the module of logarithmic vector fields in two-dimensional weighted hyperplane arrangements.

Findings

The module of logarithmic vector fields is free of rank two in the two-dimensional case.

The algorithm effectively constructs a homogeneous basis for this module.

The basis construction facilitates understanding the structure of these vector fields.

Abstract

We consider logarithmic vector fields parametrized by finite collections of weighted hyperplanes. For a finite collection of weighted hyperplanes in a two-dimensional vector space, it is known that the set of such vector fields is a free module of rank two whose basis elements are homogeneous. We give an algorithm to construct a homogeneous basis for the module.