On the Jacobian ideal and sufficiency of jets. (English) Zbl 0808.57022
Let \({\mathcal E}_ n\) be the local \(\mathbb{R}\)-algebra of germs at \(0 \in \mathbb{R}\) of smooth functions, \({\mathfrak M}_ n\) its maximal ideal. For any \(f \in {\mathcal E}_ n\), we put \(\nu(f) = \sup \{k \in \mathbb{N} \mid f \in {\mathfrak M}_ n^ k\}\). The main result in this paper is as follows:
Theorem. Let \(f \in {\mathfrak M}^ 3_ n\) \((n \geq 2)\). Suppose that there exists an integer \(\rho \geq 1\) such that \({\mathfrak M}^ \rho_ n \subset J(f)\). Then (i) \(\rho \geq n(\nu(f) - 2) + 1\), (ii) \({\mathfrak M}_ n^ \rho \subset {\mathfrak M}^{\nu (f) - 2}_ n J(f)\). Here, \(J(f)\) is the Jacobian ideal of \(f\).
The author gives some applications to the sufficiency of jets.
Theorem. Let \(f \in {\mathfrak M}^ 3_ n\) \((n \geq 2)\). Suppose that there exists an integer \(\rho \geq 1\) such that \({\mathfrak M}^ \rho_ n \subset J(f)\). Then (i) \(\rho \geq n(\nu(f) - 2) + 1\), (ii) \({\mathfrak M}_ n^ \rho \subset {\mathfrak M}^{\nu (f) - 2}_ n J(f)\). Here, \(J(f)\) is the Jacobian ideal of \(f\).
The author gives some applications to the sufficiency of jets.
Reviewer: S.Izumiya (Sapporo)
MSC:
| 57R45 | Singularities of differentiable mappings in differential topology |
| 58C25 | Differentiable maps on manifolds |
| 58K99 | Theory of singularities and catastrophe theory |
