Represent MOD function by low degree polynomial with unbounded one-sided error

In this paper, we prove tight lower bounds on the smallest degree of a nonzero polynomial in the ideal generated by $MOD_q$ or $\neg MOD_q$ in the polynomial ring $F_p[x_1, \ldots, x_n]/(x_1^2 = x_1, \ldots, x_n^2 = x_n)$, $p,q$ are coprime, which is called \emph{immunity} over $F_p$. The immunity of $MOD_q$ is lower bounded by $\lfloor (n+1)/2 \rfloor$, which is achievable when $n$ is a multiple of $2q$; the immunity of $\neg MOD_q$ is exactly $\lfloor (n+q-1)/q \rfloor$ for every $q$ and $n$. Our result improves the previous bound $\lfloor \frac{n}{2(q-1)} \rfloor$ by Green. We observe how immunity over $F_p$ is related to $\acc$ circuit lower bound. For example, if the immunity of $f$ over $F_p$ is lower bounded by $n/2 - o(\sqrt{n})$, and $|1_f| = \Omega(2^n)$, then $f$ requires $\acc$ circuit of exponential size to compute.