Exercise 111.26.10. Let $k$ be a field and let $A = k$. Let $\varphi (M) = \dim _ k(M)$. Fix $d\in {\mathbf N}$. Consider the graded $A$-algebra $B = k[x, y, z]/(x^ d + y^ d + z^ d)$, where $x, y, z$ each have degree $1$. Compute the Hilbert function of $B$. Is there a Hilbert polynomial in this case?
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