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!