Exact Divisibility of Exponential Sums Associated to Binomials over finite fields

In this paper we compute the exact divisibility of exponential sums associated to binomials $F(X)=aX^{d_1} +b X^{d_2}$. In particular, for the case where $\max\{d_1,d_2\}\leq\sqrt{p-1}$, the exact divisibility is computed. As a byproduct of our results, we obtain families of binomials that do not permute $\mathbb{F}_p$, and a lower bound for the sizes of value sets of binomials over $\mathbb{F}_p$. Additionally, we obtain a new criterion to determine if a polynomial defines or not a permutation of $\mathbb{F}_p$ that depends on the divisibility of the exponential sum associated to the polynomial.