On the polynomials x d +ax i +b and x d +ax d−i +b over $\mathbb{F}_{q}$ and Gaussian hypergeometric series

Some mixed character sum identities of Katz II

“…Hypergeometric character sums over finite fields have had a variety of applications in number theory. For some recent examples, see [2], [3], [7], [11], [14], [15], [16].…”

Quadratic Form Gauss Sums

“…The finite field Gauss sum has applications in many finite field problems. One sees their use in coding theory [37], [116]; point counting on elliptic curves [97], [117]; determining the number of solutions to polynomial equations [7], [95]; determining the value of cyclotomic numbers [114]; constructing difference sets [39]; and the evaluation of various related functions [85]. In particular, the finite field Gauss sums are widely used in the construction of Cayley graphs [38], [40], [43], [87], [88], [113], [115].…”

“…The approach by Alaca, et al is to determine integers ρ, σ, τ, µ such that, modulo p n , we Subsequently, they show that there exists an automorphism λ defined on Z p n × Z p n which is given by λ(x, y) = (ρx + σy, τ x + µy). Due to their choice of integers, by setting A = aρ 2 + bρτ + cτ 2 , we have 7) and in particular we will have (A, p) = 1. Hence, it follows that…”