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
2025-01-13 06:30:14.1736749814
Hilbert function and Hilbert polynomial
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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.
See e.g. http://tartarus.org/gareth/maths/notes/iii/Commutative_Algebra_2013.pdf, p. 31 onwards, for an approach to it.