Let J⊂I be ideals in a formally equidimensional local ring with λ(I/J)<∞. Rees proved that λ(In/Jn) 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)≤dimR−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⊂I in a formally equidimensional local ring, we consider the multiplicity function e(In/Jn) which is a polynomial in n for all large n. We explicitly determine the deg⁡e(In/Jn) in some special cases. For an ideal J of analytic deviation one, we give characterization of reductions in terms of deg⁡e(In/Jn) under some additional conditions. © 2019 Elsevier B.V.