Solution
= Solution
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
$$
P_R(z)=\frac{1-z^k}{(1-z)^2\prod_{i=1}^n(1-z^i)}.
$$
Solved by gpt-5.6-sol high.