# Rees' theorem for filtrations, multiplicity function and reduction criteria

## TL;DR

This paper extends Rees' theorem to broader classes of filtrations in local rings, linking polynomial degrees of length and multiplicity functions to reduction criteria, with explicit degree characterizations in special cases.

## Contribution

It generalizes Rees' theorem to Noetherian and non-Noetherian filtrations, providing new criteria for reductions based on polynomial degrees of length and multiplicity functions.

## Key findings

- Degree of length polynomial indicates reduction status.
- Explicit degree formulas for multiplicity functions in certain cases.
- Characterization of reductions via polynomial degrees for ideals with analytic deviation one.

## Abstract

Let $J\subset I$ be ideals in a formally equidimensional local ring with $\lambda(I/J)<\infty.$ Rees proved that for all $n\gg0$, $\lambda(I^n/J^n)$ is a polynomial $P(I/J)(X)$ in $n$ of degree at most dim $R$ and $J$ is a reduction of $I$ if and only if deg $P(I/J)(X)\leq$ dim $R-1.$ We extend this result for all Noetherian filtrations of ideals in a formally equidimensional local ring and for (not necessarily Noetherian) filtrations of ideals in analytically irreducible rings. We provide certain classes of ideals such that deg $P(I/J)$ achieves its maximal degree. On the other hand, for ideals $J\subset I$ in a formally equidimensional local ring, we consider the multiplicity function $e(I^n/J^n)$ which is a polynomial in $n$ for all large $n.$ We explicitly determine the deg $e(I^n/J^n)$ in some special cases. For an ideal $J$ of analytic deviation one, we give characterization of reductions in terms of deg $e(I^n/J^n)$ under some additional conditions.