Near-exceptionality over infinite fields

Polynomial and rational maps over finite fields which are bijective have been widely studied. We study rational maps which are almost surjective. We describe a family of rational maps over finite fields which are very close to surjective. We define the family in terms of a quantity from Galois Theory which gives an approximation to the size of the image for the rational functions. We coin the term “near-exceptional” to describe rational functions in the family in analogy to the term “exceptional” which refers to a family of rational functions that are bijective.; We define near-exceptionality. We classify near-exceptional rational functions over finite fields and we give the size of the image for these maps. We describe some properties common to near-exceptional rational functions.