On the number of distinct values of a class of functions over a finite field

Several authors have recently shown that a planar function over a finite field of order q must have at least (q+1)/2 distinct values. In this note this result is extended by weakening the hypothesis significantly and strengthening the conclusion. We also give an algorithm for determining whether a given bivariate polynomial ϕ(X,Y) can be written as f(X+Y)−f(X)−f(Y) for some polynomial f. Using the ideas of the algorithm, we then show a Dembowski–Ostrom polynomial is planar over a finite field of order q if and only if it yields exactly (q+1)/2 distinct values under evaluation; that is, it meets the lower bound of the image size of a planar function.