Further reductions of Poincaré-Dulac normal forms in {{????}}^{{}????+1}

In this paper, we will consider (germs of) holomorphic mappings of the form ( f ( z ) , λ 1 w 1 ( 1 + g 1 ( z ) ) , … , λ n w n ( 1 + g n ( z ) ) ) (f(z),\lambda _{1} w_{1}(1+g_{1}(z)),\ldots ,\lambda _{n}w_{n}(1+g_{n}(z))) , defined in a neighborhood of the origin in C n + 1 {\mathbf {C}}^{n+1} . Most of our interest is in those mappings where f ( z ) = z + a m z m + ⋯ f(z)=z+a_{m}z^{m}+\cdots is a germ tangent to the identity and g i ( 0 ) = 0 g_{i}(0)=0 for i = 1 , … , n i=1,\ldots ,n , and λ i ∈ C \lambda _{i}\in {\mathbf {C}} possess no resonances, for these are the so-called Poincaré-Dulac normal forms of the mappings ( z + O ( 2 ) , λ 1 w + O ( 2 ) , … , λ n w + O ( 2 ) ) (z+O(2), \lambda _{1}w+O(2),\ldots ,\lambda _{n}w+O(2)) . We construct formal normal forms for these mappings and discuss a condition which tests for the convergence or divergence of the conjugating maps, giving specific examples.