At every $n\in\mathbb N$ (all polynomials are of degree $O(1)$) is there $g_{3,1}^{(n)},\dots,g_{3,k}^{(n)}\in\mathbb F_3[x_1,\dots,x_n]$ at $k=\mathsf{poly}(n)$ and $g_2^{(n)}\in\mathbb F_2[x_1,\dots,x_k]$ satisfying ($g_i=g_i^{(n)}$ below) $\forall(x_1,\dots,x_n)\in\{-1,0,+1\}^n$ $$g_2\big({g_{3,1}^2(x_1,\dots,x_n)},\dots,g_{3,k}^2(x_1,\dots,x_n)\big)=0\in\mathbb F_2\iff\sum_{i=1}^nx_i=0\in\mathbb Z?$$
I associate
$2=-1$ in $\mathbb F_3$
$1^2,-1^2\in\mathbb F_3$ to $1\in\mathbb F_2$.
