The weighted <Hilbert series> of $K[x,y,t_1,\ldots,t_n]$ is
$$
\frac1{(1-z)^2\prod_{i=1}^n(1-z^i)}.
$$
The homogeneous polynomial $x^k-y^k$ has degree $k$ and is a <non-zero-divisor>, so quotienting by it multiplies the series by $1-z^k$. Therefore the requested <Poincare series of a graded module> is
$$
P_R(z)=\frac{1-z^k}{(1-z)^2\prod_{i=1}^n(1-z^i)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 101 / 5 / b / Solution

{c}

For a chosen additive function such as module length, the Poincare series of a graded module is the generating function of that function on its homogeneous components. In this usage it is a Hilbert series.


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