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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$I$) $\bar{v}_I$ $:$ $A\longrightarrow \mathbb{R}\cup {+\infty}$ is defined by$\bar{v}_I(x)=lim_{k \to \+infty}\frac{ord_I(x^k}{k}$, $\forall x \in A$.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 SamuelFunction (with respect to $I$) when passing to hyperplanes sections of $A$ asone does for the theory of mixed multiplicities.

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