Multiplicity of iterated Jacobian extensions of weighted homogeneous map germs
The authors define an invariant \(e_{\mathbf i}(f)\) called the algebraic multiplicity of the analytic map germ \(f:(\mathbb{C}^n,0) \to(\mathbb{C}^p,0)\) with respect to the Boardman symbol \({\mathbf i}=(i_1,\dots,i_k)\). This invariant is the multiplicity (in the Hilbert-Samuel sense) of the local ring \({\mathcal O}_n/J_{\mathbf i}(f)\) where \(J_{\mathbf i}(f)\) is an iterated jacobian ideal defined by \(f\), it generalizes other well-known invariants like the Milnor number. When \({\mathcal O}_n/J_{\mathbf i}(f)\) has dimension 0, \(e_{ \mathbf i}(f)\) is the colength of \(J_{\mathbf i}(f)\) in \({\mathcal O}_n\) and coincides with the invariant previously defined by the second author and \textit{M. J. Saia} in [Glasg. Math. J. 40, No. 1, 21-32 (1998; Zbl 0919.58010)]. They give some formulae for this invariant when \(f\) is weighted homogeneous. The computation is based on some results on the multiplicity of determinantal rings that they prove using the Eagon-Northcott complex.
scientific article; zbMATH DE number 1532964
scientific article