# Monomial generators of complete planar ideals

## TL;DR

This paper introduces an algorithm to compute monomial generators for complete ideals in smooth complex surfaces, providing an invariant and a geometric method for integral closure computation.

## Contribution

It presents a novel algorithm that produces monomial generators as an invariant, linking algebraic and geometric aspects of complete planar ideals.

## Key findings

- Algorithm computes generators as monomials in maximal contact elements.
- Monomial expression is an equisingularity invariant.
- Method applies to families of complete ideals.

## Abstract

We provide an algorithm that computes a set of generators for any complete ideal in a smooth complex surface. More interestingly, these generators admit a presentation as monomials in a set of maximal contact elements associated to the minimal log-resolution of the ideal. Furthermore, the monomial expression given by our method is an equisingularity invariant of the ideal. As an outcome, we provide a geometric method to compute the integral closure of a planar ideal and we apply our algorithm to some families of complete ideals.