Integral closure and bounds for quotients of multiplicities of monomial ideals
The author studies the multiplicity (and related notions) of ideals in the ring \(\mathcal{O}_{n}\) of complex analytic function germs \(f:(\mathbb{C} ^{n},0)\rightarrow \mathbb{C}\). In particular: monomial ideals, mixed multiplicities (for systems of ideals), Newton filtration of \(\mathcal{O}_{n} \) and integral closure of ideals. According to the author the main result is: for a given pair of monomials ideals \(I\) and \(J\) of finite colength in \( \mathcal{O}_{n}\) some power of \(I\) admits a reduction formed by homogeneous polynomials with respect to the Newton filtration induced by \(J\) if and only if the quotient of multiplicities \(e(I)/e(J)\) attains a suitable upper bound expressed in terms of the Newton polyhedra of \(I\) and \(J\).
scientific article; zbMATH DE number 6845990
scientific article