Integral closure and bounds for quotients of multiplicities of monomial ideals

Abstract

[EN] Given a pair of monomial ideals $I$ and $J$ of finite colength of the ring of analytic function germs $(\C^n,0)\to \C$, we prove that 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 $J$. We also explore other connections between mixed multiplicities, Newton filtrations and the integral closure of ideals.