On the Saito basis for plane curves
Emilio de Carvalho, Percy Fernández-Sánchez, Marcelo Escudeiro Hernandes
TL;DR
The paper develops and implements a practical framework to compute the Saito module $S(f)$ and its Saito basis for plane curve germs, linking analytic invariants such as the Tjurina number $\tau(f)$ to the cofactor structure via the Jacobian $J(f)$. By leveraging a standard-basis approach for the Kähler differential module and syzygy techniques, it provides algorithms to extract a Saito basis and compute $\tau(f)$, $\mu(f)$, and related data efficiently. The authors apply the method to curves with multiplicity up to $3$, deriving explicit Saito bases and closed-form expressions for $\tau(f)$ in all multiplicity cases (1, 2, and 3) and for both irreducible and reducible curves, including quasi-homogeneous examples. This work tightens the connection between the Saito module, residues, and analytic invariants, with implications for analytic classification of plane curves and foliation theory. Overall, the results offer concrete computational tools for obtaining analytic invariants and understanding the foliations that leave a plane curve invariant.$
Abstract
We present some results concerning the Saito module and the torsion submodule of an analytic plane curve, and we provide a method for computing them. Using this algorithm, we compute analytic invariants for plane curves with multiplicity less than or equal to three.