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Let $k$ be a finite field. Let $P_r$ be set of degree $r$ binary forms over $k$. We define $$ \mathcal{M}_r = \{ (x_0, ..., x_n) \in P_r^{n+1} : x_0^d + ... + x_n^d = 0 \text{ and the }x_i \text{'s don't have a common factor} \}. $$

I am interested in knowing what the value $$ \#\mathcal{M}_r/\#\mathcal{M}_{r-d} $$ is when $d<r.$

The reason why I am asking this is because I am reading a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, and it seems to be using this value in the proof of Lemma 2.4.4 on page 28.

I would greatly appreciate any help to figure this out. Thanks!

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