Exact Divisibility of Exponential Sums Associated to Binomials over finite fields

Abstract

In this paper we compute the exact divisibility of exponential sums associated to binomials $F(X)=a{X}^{d_1}+b{X}^{d_2}$. In particular, for the case where $\max \{d_1,d_2\}\le \sqrt{p-1}$, 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.

arXiv commentThis paper has been withdrawn by the author due to a crucial sign error in equation 1