Numerical characterizations for integral dependence of graded ideals

Let $R=\oplus_{m\geq 0}R_m$ be a standard graded Noetherian domain over a field $R_0$ and $I\subseteq J$ be two graded ideals in $R$ such that $0<\mbox{height}\;I\leq \mbox{height}\;J {\bf d}$. The statement $(2)$ generalizes the classical result of Rees. The statement $(3)$ gives the integral dependence criteria in terms of the Hilbert-Samuel multiplicities of certain standard graded domains over $R_0$. As a consequence of $(3)$, we also get an equivalent statement in terms of (Teissier) mixed multiplicities. Apart from several well-established results, the proofs of these results use the theory of density functions which was developed recently by the authors.