Fonction asymptotique de Samuel des sections hyperplanes et multiplicité

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Let (A,mₐ,k) be a local noetherian ring and I an mₐ-primary ideal. The asymptotic Samuel function (with respect to I) vᵢ: A R∪ +∞ is defined by vᵢ(x)=limk → inftyordᵢ(xᵏ/k, ∀ x ∈ A. Similary, one defines for another ideal J, vᵢ(J) as the minimum of vᵢ(x) as x varies in J. Of special interest is the rational number vᵢ(mₐ). We study the behavior of the Asymptotic Samuel Function (with respect to I) when passing to hyperplanes sections of A as one does for the theory of mixed multiplicities.