On a conjecture of Azevedo (Q1805498)

Let \({\mathfrak o}\) be an irreducible plane algebroid curve over an algebraically closed field \(k\) of characteristic zero. By \(\widetilde \Omega ({\mathfrak o}/k)\) we denote the universally finite differential module of \({\mathfrak o}\) over \(k\). In his Ph. D. thesis ``The Jacobian ideal of a plane algebroid curve'' (Purdue Univ. 1967), \textit{A. de Azevedo} conjectured that for a given equisingularity class \({\mathcal E}\) of irreducible plane algebroid curves, the canonical branch of \({\mathcal E}\) has the biggest length of the torsion submodule of \(\widetilde \Omega ({\mathfrak o}/k)\) (where \({\mathfrak o} \in {\mathcal E})\). In this article an algorithm is developed to compute the length of the torsion submodule for any curve \({\mathfrak o}\) given by a parametrization. A counterexample to Azevedo's conjecture is given.