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

Table of contents
- Solution b

Solution

The weighted Hilbert series of

K [x, y, t_{1}, \dots, t_{n}]

is

\frac{1}{( 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.

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