# Canonical bases of modules over one dimensional k-algebras

## TL;DR

This paper introduces a method to compute the degree monoid of modules over polynomial algebras and applies it to classify plane algebraic curves based on invariants derived from Kähler differentials.

## Contribution

It provides a novel procedure for calculating degree monoids of modules over one-dimensional k-algebras and demonstrates applications to classifying polynomial curves using algebraic invariants.

## Key findings

- Procedure to compute monoid of degrees for modules over K-algebras.
- Application to classification of plane polynomial curves.
- Use of Kähler differentials for invariant-based classification.

## Abstract

Let K be a field and denote by K[t], the polynomial ring with coefficients in K. Set A = K[f1,. .. , fs], with f1,. .. , fs $\in$ K[t]. We give a procedure to calculate the monoid of degrees of the K algebra M = F1A + $\times$ $\times$ $\times$ + FrA with F1,. .. , Fr $\in$ K[t]. We show some applications to the problem of the classification of plane polynomial curves (that is, plane algebraic curves parametrized by polynomials) with respect to some oh their invariants, using the module of K{\"a}hler differentials.