Effective calculations of the multiplicity of polynomial mappings

In this paper we give an effective characterization of the set of linear mappings (L1,…,Ln):Cm→Cn(L1,…,Ln):Cm→Cn for which the multiplicity of a polynomial mapping F:Cn→CmF:Cn→Cm at an isolated zero a∈Cna∈Cn is equal to the codimension of the ideal (L1∘F,…,Ln∘F)(L1∘F,…,Ln∘F) in the local ring Oan of holomorphic functions germs at a∈Cna∈Cn.