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Let $J\subset I$ be ideals in a formally equidimensional local ring with$\lambda(I/J)<\infty.$ Rees proved that for all $n\gg0$, $\lambda(I^n/J^n)$ isa polynomial $P(I/J)(X)$ in $n$ of degree at most dim $R$ and $J$ is areduction of $I$ if and only if deg $P(I/J)(X)\leq$ dim $R-1.$ We extend thisresult for all Noetherian filtrations of ideals in a formally equidimensionallocal ring and for (not necessarily Noetherian) filtrations of ideals inanalytically irreducible rings. We provide certain classes of ideals such thatdeg $P(I/J)$ achieves its maximal degree. On the other hand, for ideals$J\subset I$ in a formally equidimensional local ring, we consider themultiplicity 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 anideal $J$ of analytic deviation one, we give characterization of reductions interms of deg $e(I^n/J^n)$ under some additional conditions.

Rees' theorem for filtrations, multiplicity function and reduction criteria