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The relation between-the determinacy order of a given germ and the length of its Boardman sequence was one of the main problems which was considered in [14]. We have concluded later that better results can be obtained if a stronger formulation for finite determinacy is used. This suggested us the definition of (r,k)-determination and its further characterization, in Chapter II. We have also studied in the same Chapter, the stability under small perturbations of this new concept and we registered several mistakes which were found in the related literature. The central problem in Chapter III is the following: "Given a germ f : Kⁿ,0 → K,0 when is it possible to find coordinate systems in such a way that f has the form: f(x,y) = g(x) + h(y) , x = (x₁,...,x_r) and y = (x_r+1, ...X_n) ? Several differentiable invariants are also suggested and by using them, it is possible to give necessary and sufficient conditions to some classes of germs. These same invariants are used in Chapter IV to prove that: "Given a germ f : Kⁿ,0 → K,0, with f ∈ m²_n and m^α_n ⊂ m^β_n< αf>, which satisfies condition D_p, then the (α-β+1)-th index of the Boardman sequence of f vanishes." , It is also given a conjecture which easily implies that. for a (r,k)-determined germ f , the (k-r+1)-th index of its Boardman sequence is zero. In Chapter V, it is given a class of polynomium in two variables, whose Boardman sequence-length is lower than the given degree. It is also given a class of polynomium in n-variables and a function h : N → N such that each polynomium f in that class satisfies the following inequality: Ι B_f Ι ≤ h (deg f).