Minimal value set binomials and Frobenius nonclassical curves

In this paper, we characterize all minimal value set binomials over ${\mathbb{F}}_q$, that is, binomials whose size of the set of images is the smallest possible. With this information, we also classify all quadrinomial curves with separated variables that are ${\mathbb{F}}_q$-Frobenius nonclassical for the morphism of lines.