Zeros of separated-variable polynomials over finite fields

Let F q \mathbb F_q be the finite field of characteristic p p with q = p a q=p^a elements. Let X X be a set of n n variables, and let A A be a nonempty subset of { 1 , … , n } \{1,\ldots ,n\} . Let N \mathbb {N} denote the set of nonnegative integers, and let σ ( d ) \sigma (d) be the digital sum of d ∈ N d\in \mathbb N in base p p . For f ( X ) = ∑ j ∈ J a j X j f(X)=\sum _{j\in J}a_{j}X^{j} with J J