Let $q$ be a power of 2. Let $P$ be the set of polynomials in $F_q [x]$ of degree d or less. Let $Z$ be the ring of integers. For any $f \in P$, let $\psi(f)$ be the number of distinct roots of $f$ in $F_q$. Note that $\psi(0) = q$.

For any map $A$ from $P$ to $Z$, one can compute the summation $$\sum_{f, g \in P} A(f)A(g) \psi (f+g).$$ My question is: what is the minimum positive value of the summation?

For $d=0,1$, the minimum is $q$. What happens if $d$ is bigger? I am especially interested in the case when $d = q/2 -1$.

Thanks a lot,

Qi