I've already asked this question on mathoverflow, but no one answered. So I put this problem also here. Sorry for that.
Let $\mathbb F_q$ be a finite field and let $e, k$ be positive integers with $k \leq e < q-1$. Define $T_e$ to be the set $$T_e:=\left\{p(x) \in \mathbb F_q[x] \;|\; \deg p =e \;, lc(p)=1, \; p(x)|x^{q-1}-1 \right\}. $$ I would like to find the value $$ M(e,k,q):=\max\left\{ | W \cap T_e| \;\; | \; W \mbox{ is a subspace of } \mathbb F_q[x]_{\leq e}, \; \dim W=k+1\right\}. $$
I could determine exactly the value $ M(e,k,q) $ in three simple cases.
When $e=k$ then we have $$M(e,e,q)= \binom{q-1}{e}.$$ In fact, since $e=k$, we can take $W=\mathbb F_q[x]_{\leq e}$ and so we obtain $W\cap T_e=T_e$.
When $k=1$, we have $$M(e,1,q)=q-e.$$
If $e=q-2$ then $$M(q-2, k, q) =k+1.$$
For the general case I conjecture that the maximum $M(e,k,q)$ is obtained by taking a polynomial $p(x)$ of degree $e-k$ that divides $x^{q-1}-1$ and choosing the subspace $$ W= \langle p(x), xp(x), \ldots, x^kp(x) \rangle.$$ From this I would obtain the following conjecture.
Conjecture: Let $q$ be a power of a prime, and let $e,k$ positive integers such that $ 0<k\leq e < q-1$. Then $$ M(e,k,q) = \dbinom{q-1-e+k}{k}.$$
Could anyone help me to prove or disprove it? Thanks in advance.
The identical mathoverflow crosspost has been answered by Will Sawin and accepted by Sfarla, see https://mathoverflow.net/q/256292