A probabilistic approach to value sets of polynomials over finite fields

In this paper we study the distribution of the size of the value set for a random polynomial with a prescribed index ℓ|(q−1) over a finite field Fq, through the study of a random r-th order cyclotomic mapping with index ℓ. We obtain the exact probability distribution of the value set size and show that the number of missing cosets (values) tends to a normal distribution as ℓ goes to infinity. A variation on the size of the union of some random sets is also considered.