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

In summary, the first step in solving a linear algebra problem is to identify the variables and equations involved. To determine if a problem has a unique solution, the number of equations must equal the number of variables and the equations must be linearly independent. A system of linear equations involves multiple equations and variables, while a matrix equation involves matrices and can be solved using matrix operations. The method used to solve a linear algebra problem depends on the type of problem and equations, and calculators and computers can be used in the process but understanding the concepts and double-checking the solutions manually is recommended.
  • #1
Ganesh Ujwal
56
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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?
 
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  • #2


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!
 

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

What is the first step in solving a linear algebra problem?

The first step is to identify the variables and equations involved in the problem. This will help determine the type of linear algebra problem and which methods and techniques can be used to solve it.

How can I determine if a linear algebra problem has a unique solution?

A linear algebra problem has a unique solution if the number of equations is equal to the number of variables and the equations are linearly independent. This means that none of the equations can be obtained by adding or multiplying other equations. If these conditions are met, then the problem has a unique solution.

What is the difference between a system of linear equations and a matrix equation?

A system of linear equations is a set of equations that involve multiple variables and can be solved simultaneously. A matrix equation, on the other hand, is a single equation that involves matrices and can be solved using matrix operations. A system of linear equations can also be represented as a matrix equation.

How do I know which method to use to solve a linear algebra problem?

The method used to solve a linear algebra problem depends on the type of problem and the given equations. Some common methods include Gaussian elimination, Cramer's rule, and matrix operations. It is important to fully understand the given problem and equations before choosing a method to solve it.

Can I use a calculator or computer to solve a linear algebra problem?

Yes, calculators and computers can be used to solve linear algebra problems. However, it is important to understand the concepts and methods behind the solutions rather than solely relying on technology. Additionally, it is always a good idea to double check the solution manually to ensure accuracy.

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