How to determine a curve singularity

Abstract

We characterize the finite codimension sub- ${\mathbf {k}}$ -algebras of ${\mathbf {k}}[\![t]\!]$ as the solutions of a computable finite family of higher differential operators. For this end, we establish a duality between such a sub-algebras and the finite codimension ${\mathbf {k}}$ -vector spaces of ${\mathbf {k}}[u]$ , this ring acts on ${\mathbf {k}}[\![t]\!]$ by differentiation.