# Hilbert function and Hilbert polynomial

I have largely studied Hilbert function and Hilbert polynomial for polynomial rings over fields of characteristic zero. Is it possible to extend the theory also for polynomial rings over fields of characteristic positive?

Thank you

Yes: for a finitely generated $k$-algebra, for instance, you can still take your additive function $\lambda$ to be $\dim_k$, and you get that the Hilbert series $\mathcal P(V,t) = \sum_{i \geq 0} \lambda(V_i)t^i$ is a rational function.
• @Ella Is the question, perhaps, about the ring where the Hilbert polynomial itself resides? Those dimensions are non-negative integers, so it is natural to view $\mathcal{P}(V,t)$ as an element of the ring of formal power series $\Bbb{Z}[[t]]\subset K[[t]]$ for any field $K$ of characteristic zero. Irrespective of the field of definition of $V$. Or,...? – Jyrki Lahtonen May 22 '16 at 7:33