A note on Ratliff-Rush filtration, reduction number and postulation number of m-primary ideals

Let $(R,m)$ be a Cohen-Macaulay local ring of dimension $d\ge 2$, and I an $m$-primary ideal. Let $r\left(I\right)$ be the reduction number of I, $n\left(I\right)$ the postulation number and $\rho \left(I\right)$ the stability index of the Ratliff-Rush filtration with respect to I. We prove that for $d=2$, if $n\left(I\right)=\rho \left(I\right)-1$, then $r\left(I\right)\le n\left(I\right)+2$, and if $n\left(I\right)\ne \rho \left(I\right)-1$, then $r\left(I\right)\ge n\left(I\right)+2$. For $d\ge 3$, assuming I is integrally closed, $\mathrm{depth}\mathrm{gr}\left(I\right)=d-2$, and $n\left(I\right)=-(d-3)$, we prove that $r\left(I\right)\ge n\left(I\right)+d$. Our main result generalizes a result by Marley on the relation between the Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the condition on the depth of the associated graded ring to the good behavior of the Ratliff-Rush filtration with respect to I mod a superficial sequence. From this result, it follows that for Cohen-Macaulay local rings of dimension $d\ge 2$, if $P_I\left(k\right)=H_I\left(k\right)$ for some $k\ge \rho \left(I\right)$, then $P_I\left(n\right)=H_I\left(n\right)$ for all $n\ge k$.