Joint reductions of monomial ideals and multiplicity of complex analytic maps

We characterize the joint reductions of a set of monomial ideals in the ring $\O_n$ of complex analytic functions defined in a neighbourhood of the origin in $\C^n$. We also study an integer $\sigma(I_1,\dots, I_n)$ attached to a family of ideals $I_1,\dots, I_n$ in a Noetherian local ring that extends the usual notion of mixed multiplicity. If $I_1,\dots, I_n$ are monomial ideals of $\O_n$, then we obtain a characterization of the families $g_1,\dots, g_n$ such that $g_i\in I_i$, for all $i=1,\dots, n$, and that $e(g_1,\dots, g_n)=\sigma(I_1,\dots, I_n)$.

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Published 1 January 2008