Fonction asymptotique de Samuel des sections hyperplanes et multiplicité

Let (A,mA,k)(A,mA,k) be a local noetherian ring and II an mAmA-primary ideal. The asymptotic Samuel function (with respect to II) vI¯: A⟶R∪{+∞}A⟶R∪{+∞} is defined by vI¯(x)=limk→+∞ordI(xk)k, ∀x∈A∀x∈A. Similarly, one defines, for another ideal JJ, vI¯(J) as the minimum of vI¯(x) as xx varies in JJ. Of special interest is the rational number vI¯(mA). We study the behavior of the asymptotic Samuel function (with respect to II) when passing to hyperplane sections of AA as one does for the theory of mixed multiplicities.