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.

中文翻译：

---

如何确定曲线奇点

我们将${\mathbf {k}} [\![t]\!]$的有限余维子${\mathbf {k}}$ 代数表征为可计算的有限高阶微分算子族的解。为此，我们在这样的子代数和有限余维${\mathbf {k}}$ -向量空间${\mathbf {k}}[u]$之间建立对偶性，该环作用于${ \mathbf {k}}[\![t]\!]$通过微分。