Joint reductions of monomial ideals and multiplicity of complex analytic maps
Let \((R, \mathfrak m)\) denote an \(n\)-dimensional local ring. For a certain family of ideals \(I_1,\ldots,I_n\) the author introduces an integer \(\sigma(I_1,\ldots,I_n)\) that extends the notion of mixed multiplicity of ideals of finite colength introduced by \textit{B. Teissier} and Risler, see [Astérisque, No. 7--8, 285--362 (1973; Zbl 0295.14003)]. It is shown by the author that it is equal to a multiplicity defined by \textit{D. Rees} [Lectures on the asymptotic theory of ideals. London Mathematical Society Lecture Note Series, 113. Cambridge: Cambridge University Press (1988; Zbl 0669.13001)]. Let \({\mathcal O}_n\) denote the ring of analytic function germs \(f : (\mathbb C^n,0) \to \mathbb C.\) Let \(I_1,\ldots,I_n\) be monomial ideals in \({\mathcal O}_n.\) There is a combinatorial characterization of of the joint reductions of \(I_1,\ldots,I_n.\) Under the assumption of \(\sigma(I_1,\ldots,I_n) < \infty\) the author proves a characterization of those analytic maps \(g = (g_1,\ldots,g_n) : (\mathbb C^n, 0) \to (\mathbb C^n, 0)\) such that \(g_i \in I_i, i = 1,\ldots,n,\) and such that \(e(g_1,\ldots,g_n) = \sigma(I_1,\ldots,I_n),\) where \(e(I_1,\ldots,I_n)\) denotes the Hilbert-Samuel multiplicity of the ideal \((g_1,\ldots,g_n)\) in \({\mathcal O}_n.\) This characterization is expressed via the respective Newton polyhedra of the \(I_i'\)s. The results are applied for the computation of the integral closure of certain ideals.
scientific article; zbMATH DE number 5310640