Fonction asymptotique de Samuel des sections hyperplanes et multiplicit\'e

Let $(A,{\mathfrak{m}}_A,k)$ be a local noetherian ring and $I$ an ${\mathfrak{m}}_A$-primary ideal. The asymptotic Samuel function (with respect to $I$) ${\bar{v}}_I$ $:$ $A⟶\mathbb{R}\cup +\mathrm{\infty }$ is defined by ${\bar{v}}_I(x)=li{m}_{k\to \text{+}infty}\frac{or{d}_I(x^k}{k}$, $\mathrm{\forall }x\in A$. Similary, one defines for another ideal $J$, ${\bar{v}}_I(J)$ as the minimum of ${\bar{v}}_I(x)$ as $x$ varies in $J$. Of special interest is the rational number ${\bar{v}}_I({\mathfrak{m}}_A)$. 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.