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

A sufficient condition is that $A$ be a standard graded algebra: it is generated as a $k$-algebra by finitely many elements of degree one. The Hilbert-Serre theorem then makes $P_A(t)$ a rational function whose denominator, after cancellation, is a power of $1-t$. When the eventual Hilbert polynomial is nonzero,where the right side uses the order of the pole at $t=1$.