Let $\mathbb{F}_q$ be a finite field where $q$ a prime, and consider a multi-variate polynomial in $\mathbb{F}_q[X_1, X_2, \dots, X_\ell]$ of the following form:
$\begin{align}\tag{1} f(X_1, \dots, X_\ell) =\sum_{a_i \in \{0, 1\}, i \in [\ell]}c_{a_1, \dots, a_\ell}X_1^{a_1}X_2^{a_2}\dots X_\ell^{a_\ell}. \end{align}$
Motivation
When $f(X_1,\dots,X_\ell)$ is expressed as in the above form, we need to store $c_{a_1, \dots, a_\ell}$ for all $(a_1, \dots, a_\ell)\in \{ 0, 1\}^\ell$, which would require $O(2^\ell \log q)$ bits in the worst case. However, for every $f(X_1, \dots, X_\ell) \in \mathbb{F}_q[X_1, X_2, \dots, X_\ell]$, if we were able to factor $f(X_1,\dots,X_\ell)$ as
$\begin{align}\tag{2} f(X_1, \dots, X_\ell) =c_0 \cdot \prod_{i \in [\ell]}(X_i + c_i ), \end{align}$
then we would only require $O(\ell \log q)$ bits by storing only $c_0$ and $\{c_i\}_{i \in [\ell]}$. In fact, it would still be useful if the above can be expressed as
$\begin{align}\tag{3} f(X_1, \dots, X_\ell) = \sum_{t = [L]}c_{t,0} \cdot \prod_{i \in [\ell]}(X_i + c_{t,i}), \end{align}$
for a small enough $L$, such as $L=2$ or $L=O(\log \ell)$. In this case we would require $O(\ell L \log q)$ bits.
Question
For every multi-variate polynomial in $\mathbb{F}_q[X_1, X_2, \dots, X_\ell]$ with the restriction that each degree is at most 1, can we express it as Eq.(2) or Eq.(3)??
The reason behind Eq.(3) stems from the fact that, there exit univariate polynomials $f(X)$ over $\mathbb{F}_q[X]$ such that $f(X)$ does not factor into a product of degree one polynomials, e.g., $X^4+1$. Although it is not completely relevant, it seemed to me that there might exist some polynomials $f(X_1, \dots, X_\ell)$ such that it can not be expressed as Eq.(2).