How to determine a curve singularity

Abstract

We characterize the finite codimension sub-k-algebras of $\mathbf{k} \llbracket t \rrbracket$ 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 $\mathbf{k} \llbracket t \rrbracket$ by differentiation.