Let $n\geq1$. Let $p$ be prime. Let $F(x_1,...,x_n)=a_1x^{e_1}+a_2x^{e_2}+...+a_mx^{e_m}\in \mathbb{F}_p[x_1,...,x_n]$, having degree $d$. (Note: The $e_i$ terms are in $\mathbb{Z^n}$, but not nessecarily the elementary basis vectors)
Let $k_1,...k_m\in \mathbb{Z_{\geq 0}}$ such that $k_1+...+k_m=(p-1)p^n$. Let $σ(k_i)$ denote the sum of digits in the base $p$ expansion of $k_i$.
Let $s$ denote the number of nonzero coordinates in the vector $σ(k_1)e_1+...+σ(k_m)e_m$. [By "nonzero," I really mean nonzero mod p-1]
I want to show that $s(p-1)-(p-1)d \leq d(σ(k_1)+...+σ(k_m)-(p-1))$.
If I can do this, I should be able finish proving a special case of the Ax–Katz Theorem.
I noticed that $d=max\{$sum of coordinates in $ e_i : 1\leq i \leq m \}$, and I think this will be key to proving the inequality highlighted above.