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

Abstract

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 Samuel Function (with respect to $I$) when passing to hyperplanes sections of $A$ as one does for the theory of mixed multiplicities.