What Is the Maximum Size of Polynomial Subspaces in Finite Fields?

[Ganesh Ujwal]

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?

 

[jvicens]

Dear fellow scientist,

Your conjecture seems to be a promising approach to finding the maximum value of $M(e,k,q)$. However, it is always important to test conjectures with concrete examples and to provide a proof if possible. I suggest trying out some small examples with different values of $q, e,$ and $k$ to see if the conjecture holds true. If it does, then you can try to prove it using mathematical induction or other techniques.

Another suggestion is to consider the structure of the set $T_e$. Can you find any patterns or relationships that can help you determine the maximum value of $|W \cap T_e|$? This may also provide some insight into your conjecture.

Furthermore, it may be helpful to consult with other experts in the field or to do some literature review to see if there are any existing results or similar problems that can assist you with your conjecture. Collaboration and discussion can often lead to new ideas and approaches.

I wish you all the best in your research and hope that you are able to solve this problem. Keep exploring and never give up!