Numerical Characterizations for Integral Dependence of Graded Ideals

Abstract Let $R$ be an equidimensional standard graded Noetherian algebra over $k$. Let $e(R)$ denote the Hilbert–Samuel multiplicity of $R$. For any graded ideal $L$ of $R$, the Rees algebra $R[Lt]$ is a bigraded $k$-algebra. For any integer $c\geq 1$, we denote by $R[Lt]_{\Delta _{(c,1)}}$ the $(c,1)$-diagonal subalgebra of $R[Lt]$, by $\varepsilon (L)$ the $\varepsilon $-multiplicity of $L$, and by $e_{i}(R[Lt])$ the mixed multiplicities of $R[Lt]$. Let $I\subseteq J$ be two non-nilpotent graded ideals in $R$. Let $S$ be the standard graded $k$-algebra $R[y]$, where $y$ is an indeterminate with $\deg y=1$. Let $d_{I,J}$ be the maximum of the generating degrees of both $I$ and $J$. Let $\textsf{I} = IS$ and $\textsf{J} = JS$. We show that $\overline{I}=\overline{J}$ if and only if $e\big (S[\textsf{I}t]_{\Delta _{(c,1)}}\big ) = e\big (S[\textsf{J}t]_{\Delta _{(c,1)}}\big )$ for some $c \ge d_{I,J}$. If $R$ is a domain, then $\overline{I}=\overline{J \Longleftrightarrow \varepsilon (I)=\varepsilon (J)}$ and $e_{i}(R[It]) = e_{i}(R[Jt])$ for all $0\leq i \lt \dim R/I \Longleftrightarrow \varepsilon (I)=\varepsilon (J)$ and $e\big (R[It]_{\Delta _{(c,1)}}\big ) = e\big (R[Jt]_{\Delta _{(c,1)}}\big )$ for some $c \gt d_{I,J}$. Moreover, $\overline{I}=\overline{J}$ if and only if the truncated ideals $\textsf{I}_{\geq c}$ and $\textsf{J}_{\geq c}$ share the same the polar multiplicity or the $\varepsilon $-multiplicity or the $j$-multiplicity for some $c \gt d_{I,J}$. The proofs of these results rely on the theory of density functions, which was developed in [15].