Let $\mathbb F_q$ be a finite field (where $q$ is in general a power of a prime), and let $e, k$ be positive integers with $k \leq e < q-1$. Let $f_0(x), \ldots, f_k(x) \in \mathbb F_q[x]$ be polynomials such that $$e= \deg f_0 > \deg f_1 > \cdots > \deg f_k,$$ and let us define $W:= \langle f_0, f_1, \ldots, f_k \rangle$ the subspace of dimension $k+1$ of $\mathbb F_q[x]_{\leq e}$ seen as vector space over $\mathbb F_q$. Furthermore let $T_e$ be the set defined by $$T_e:=\left\{p(x) \in \mathbb F_q[x] \;|\; \deg p =e \;, lc(p)=1, \; p(x)|x^{q-1}-1 \right\}. $$ I'd like to find $$ M_{e,k,q}:=\max | W \cap T_e|, $$ where the maximum is over all the subspaces of the form described above. Three simple cases are:

When $e=k$ then $M_{e,e,q}= \binom{q-1}{e}$.

When $e=q-2$ then $M_{q-2, k, q} =k+1$.

When $k=1$ then $M_{e,1,q}=q-e$.

When $k=0$ then $M_{e,0,q}=1$.

Maybe it could be useful the fact that, for $e<q-1$, $T_e$ is a set of generators of $\mathbb F_q[x]_{\leq e}$.

I think it's a quite difficult question, but i'd just like to know some non-obvious bounds on $|W \cap T_e|$.

**EDIT:**
Of course $M_{e,k,q}\geq\dbinom{q-1-e+k}{k}$. Let us take
$$f_k(x)= \prod_{i=1}^{e-k}(x- \alpha_i),$$ where $\alpha_i \in \mathbb F_q^*$ for every $i=1, \ldots, e-k$ and $\alpha_i \neq \alpha_j$ for every $i \neq j$. Moreover let us take,
for $j=0, \ldots, k-1$,
$$ f_j(x)= x^{k-j}f_k(x).$$
Then $W:=\langle f_0, \ldots, f_k \rangle$ has exactly $\dbinom{q-1-e+k}{k}$ elements of $T_e$.

I suppose that $M_{e,k,q}$ is exactly $\dbinom{q-1-e+k}{k}$. In fact in the three cases described above it seems to work. Could anyone help me to prove it?

Thanks.