Integral closure and bounds for quotients of multiplicities of monomial ideals

Given a pair of monomial ideals I and J of finite colength of the ring of analytic function germs (Cn,0)→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.