The Hilbert function of the Ratliff–Rush filtration

The Ratliff–Rush filtration has been shown to be a very useful tool for studying numerical invariants of the associated graded ring Gcolon, equalscircled plustgreater-or-equal, slanted0(It/It+1) of a local ring image with respect to the classical I-adic filtration. The advantage of this approach is that the associated graded ring image of A with respect to the Ratliff–Rush filtration has positive depth, but unfortunately image is not necessarily a standard graded algebra. In this paper, we study some numerical invariants of image when I is an image-primary ideal of a local Cohen–Macaulay ring and, as consequence, we prove an upper bound on the first coefficient of the Hilbert polynomial of G which extends the already known bounds.