Suppose we are given the following n polynomials in $\bar{\mathbb{F}}_2[x_1,...,x_n]$:
$f_1 = x_1 + x_n^2$
$f_2 = x_2 + x_1^2$
$\cdot$
$\cdot$
$f_{n-3} = x_{n-3} + x_{n-4}^2$
$f_{n-2} = x_{n-2} + x_{n-1}^2$
$f_{n-1} = x_{n-1} + x_{n-2}^2$
$f_n = x_{n-2}\cdot x_{n-1} + x_{n-3}^2$
then there exists $A \in \bar{\mathbb{F}}_2[y_1,.., y_{n-1}]$ such that
$f_n + A(f_1,...,f_{n-1}) = x_n^{2^{n-2}} \mod{\langle x_1,...,x_n \rangle}^{2^{n-2}+1}$.
Moreover this $A$ is unique. I am interested in finding $||A|| = | supp(A) |$. Note that if $A = \sum_{e}a_e \bar{x}_n^{e}$, then $supp(A) = \{ e \in \mathbb{Z}^n | a_e \neq 0 \}$.
On solving small cases I believed $A$ consisted of terms of all degrees, but on computationally finding $A$ for slightly large $n$, it seems $supp(A) \subset C + C$ where $C = \displaystyle\bigcup_{1 \leq k \leq n} \displaystyle\bigcup_{0 \leq j \leq n} 2^j \cdot e_k$ where $e_k = (0,...,0,1,0,...,0)$ where 1 is in the $k^{th}$ position. So my question is can we prove $supp(A) \subset C + C ?$
As for the context this problem came up in my research and solving this special case might be of highly instructional value to me.
