Take $A=k[x]$ with $\deg x=2$. This is a Noetherian <graded algebra> with $A_0=k$, but
$$
\dim_kA_n=
\begin{cases}
1,&n\ \text{even},\\
0,&n\ \text{odd}.
\end{cases}
$$
No polynomial can agree eventually with these alternating values, so $A$ has no Hilbert polynomial.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 101 / 5 / c / iii / Solution

A graded algebra over a field $k$ is a $k$-algebra $A=\bigoplus_{n\geq0}A_n$ whose multiplication satisfies $A_iA_j\subseteq A_{i+j}$.


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