The deformation multiplicity of a map germ with respect to a Boardman symbol (Q2771439)

The paper deals with the germs of analytic maps \(f: ({\mathbb C}^n,0) \rightarrow ({\mathbb C}^p, 0)\). The goal is to define an analog of the multiplicity of \(f\) involving the higher derivatives. Namely let \({\mathbf i}=(i_1, \dots , i_k)\) be a Boardman symbol, \(J_{\mathbf i}(f)\) - the iterated Jacobian extension of \(f\), \(F: ({\mathbb C}^r \times {\mathbb C}^n,0) \rightarrow ({\mathbb C}^r \times {\mathbb C}^p, 0)\) an unfolding of \(f\), \({\mathcal M}_r\) and \({\mathcal M}_n\) the maximal ideals of \({\mathcal O}_r\) and \({\mathcal O}_n\). The deformation multiplicity of \(F\) with respect to \(\mathbf i\) is a multiplicity of the factor ring \({\mathcal O}_{r+n} / J_{\mathbf i}(F) \) with respect to the pair \(({\mathcal M}_r, {\mathcal M}_n)\). The main result is that under some regularity conditions on \(\mathbf i\), \(f\) and \(F\), defined in such a way multiplicity \(m_{\mathbf i}(F,f)\) is independent of the choice of \(F\).