For an odd prime p and an integer w ≥ 1, polynomial quotients $q_{p,w}\left(u\right)$ are defined by $q_{p,w}\left(u\right)\equiv (u^w-u^{wp})/pmodp$ with $0\le q_{p,w}\left(u\right)\le p-1$, u ≥ 0, which are generalizations of Fermat quotients $q_{p,p-1}\left(u\right)$. First, we estimate the number of elements $1\le u<N\le p$ for which $f\left(u\right)\equiv q_{p,w}\left(u\right)modp$ for a given polynomial f(x) over the finite field ${}_p$. In particular, for the case f(x)=x we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each element of...

Classical Kloosterman sums have a prominent role in the study of automorphic forms on GL${}_2$ and further they have numerous applications in analytic number theory. In recent years, various problems in analytic theory of automorphic forms on GL${}_3$ have been considered, in which analogous GL${}_3$-Kloosterman sums (related to the corresponding Bruhat decomposition) appear. In this note we investigate the first four power-moments of the Kloosterman sums associated with the group SL${}_3\left(\mathbb{Z}\right)$. We give formulas for the...

We consider Weil sums of binomials of the form $W_{F,d}\left(a\right)={\sum }_{x\in F}\psi (x^d-ax)$, where F is a finite field, ψ: F → ℂ is the canonical additive character, $gcd(d,|F^{\times }\left|\right)=1$, and $a\in F^{\times }$. If we fix F and d, and examine the values of $W_{F,d}\left(a\right)$ as a runs through $F^{\times }$, we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $|F^{\times }|$). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n odd, and $d=3^r+2$ with...