# Standard Bases for Fractional Ideals of the Local Ring of an Algebroid Curve

## TL;DR

This paper introduces an algorithm to compute standard bases for fractional ideals in the local ring of an algebroid curve, enabling the determination of semimodules and semirings of values, which are key invariants of the curve.

## Contribution

The paper presents a novel algorithm for computing standard bases of fractional ideals in local rings of algebroid curves, extending to the Kähler differential module for complex curves.

## Key findings

- Algorithm computes semimodules of values for fractional ideals.
- Finite generators obtained for the semiring of values of the curve.
- Application to the Kähler differential module yields important analytic invariants.

## Abstract

In this paper we present an algorithm to compute a Standard Basis for a fractional ideal $\mathcal{I}$ of the local ring $\mathcal{O}$ of an $n$-space algebroid curve with several branches. This allows us to determine the semimodule of values of $\mathcal{I}$. When $\mathcal{I}=\mathcal{O}$, we may obtain a (finite) set of generators of the semiring of values of the curve, which determines its classical semigroup. In the complex context, identifying the K\"{a}hler differential module $\Omega_{\mathcal{O}/\mathbb{C}}$ of a plane curve with a fractional ideal of $\mathcal{O}$ and applying our algorithm, we can compute the set of values of $\Omega_{\mathcal{O}/\mathbb{C}}$, which is an important analytic invariant associated to the curve.