For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P\_{k,\ell} = \prod\_{1\leq a\_1+\ldots+a\_k\leq \ell} (a\_1x\_1+\ldots + a\_kx\_k)\in \mathbb{F}\_2[x\_1,\ldots, x\_k] \end{equation}


consisting of all products of all non-zero linear forms $a\_1x\_1+\ldots +a\_kx\_k$, at most $\ell$ of whose coefficients are non-zero. 


Now let \begin{equation} \alpha\_j = \sum\_{i=0}^{\ell-1}\binom{j-1}{i}, \end{equation} 


and


\begin{equation} D\_{k,\ell}= \det \begin{pmatrix} x\_1^{\alpha\_1} & \cdots & x\_k^{\alpha\_1} \\
\vdots & \vdots & \vdots \\
x\_1^{\alpha\_k} & \cdots & x\_k^{\alpha\_k} \end{pmatrix} = \sum\_{\sigma \in S\_k} x\_{\sigma(1)}^{\alpha\_1} x\_{\sigma(2)}^{\alpha\_2}\cdots x\_{\sigma(k)}^{\alpha\_k}, \end{equation} 


where $S\_k$ is the symmetric group.


(Note that $\alpha\_{1}=1$, and we assume as usual that $\binom{a}{b}=0$ if $a<b$.) 


**Question**: It is a classical fact that $D\_{k,\ell}=P\_{k,\ell}$ when $\ell =2, k$. Is it true that, for arbitrary $2\leq \ell \leq k$, $D\_{k,\ell}$ is $P\_{k,\ell}$ plus (possibly) some additional terms? 



Dickson/determinant type polynomial (updated)
Is this true? Sage disagrees:



```
P.<a,b,c,d> = PolynomialRing(GF(2))
xs = P.gens()
M = Matrix(P, [[xs[i] ** (1 + binomial(j, 1) + binomial(j,2)) for j in range(4)] for i in range(4)])
M.determinant().factor()

```

returns



```
d * c * (c + d) * b * (b + d) * (b + c) * a * (a + d) * (a + c) * (a + b) * (a^3*b + a^2*b^2 + a*b^3 + a^3*c + b^3*c + a^2*c^2 + b^2*c^2 + a*c^3 + b*c^3 + a^3*d + b^3*d + a*b*c*d + c^3*d + a^2*d^2 + b^2*d^2 + c^2*d^2 + a*d^3 + b*d^3 + c*d^3)

```

This has all the right factors with $1$ or $2$ addends (which is no surprise), but the remaining degree-$4$ factor does not factor further as you want it to. For comparison, the product of the factors with $3$ addends should be



```
P.prod([P.sum(xs)-r for r in xs])

```

which is



```
a^3*b + a*b^3 + a^3*c + b^3*c + a*c^3 + b*c^3 + a^3*d + b^3*d + a*b*c*d + c^3*d + a*d^3 + b*d^3 + c*d^3

```

Generally, I think you can learn something about these polynomials from: [I. G. Macdonald, *Schur functions: Theme and variations*, Séminaire Lotharingien de Combinatoire 1992, volume 28, page B28a](https://eudml.org/doc/119412) (specifically, the 7th variation, which does not assume much familiarity with the preceding ones). But do not expect explicit formulas beyond the $\ell = 2$ and $\ell = k$ cases.


