The local invariant factors of a product of holomorphic matrix functions: The order 4 case

Abstract

Let$\lambda ^{\alpha _n } |\lambda ^{\alpha _{n - 1} } |...|\lambda ^{\alpha _2 } |\lambda ^{\alpha _1 }$, resp.$\lambda ^{\beta _n } |\lambda ^{\beta _{n - 1} } |...|\lambda ^{\beta _2 } |\lambda ^{\beta _1 }$ be the (given) invariant factors of the square matricesA, resp.B of ordern over the ring of germs of holomorphic functions in 0 such that detA(λ)B(λ)≠0, λ≠0. A description of all possible invariant factors$\lambda ^{\gamma _n } |\lambda ^{\gamma _{n - 1} } |...|\lambda ^{\gamma _2 } |\lambda ^{\gamma _1 }$ of the productC=AB is given in the following cases: (i)β ₁ (or α₁)≤2; (ii)β ₃ = 0 (α₃= 0); (iii) α₁−α₂, β₁β _m ≤1,α ₂₊₁−β _m+1−0. These results, which hold for arbitraryn, are complemented with a few results leading to the description of all possible exponents γ₁,γ₂,γ₃,γ₄ for arbitrary α₁,α₂,α₃,α₄ β₁,β₂,β₃,β₄ in the case where the ordern≤4.