Let $k$ be an algebraicaly closed field of characteristic $p>0$. Let $p_1,p_2\in k[x_1,\dots,x_n]$ be polynomials and let ${\rm V}(p_1,p_2)\subset k^n$ be the set of zeros of both $p_1$ and $p_2$.
Question. If ${\rm V}(p_1,p_2)$ is an additive group, are there additive polynomials $\pi_1,\dots,\pi_m$, (ie satisfying $\pi_i(\bar x+\bar y)=\pi_i(\bar x)+\pi_i(\bar y)$) such that ${\rm V}(p_1,p_2)={\rm V}(\pi_1,\dots,\pi_m)$ ?
It seems to be known that if ${\rm V}(p_1)$ is an additive subgroup of $k^n$, and $p_1$ a square-free polynomial, then $p_1$ is additive indeed.