Explicit formulae for the number of triplets of consecutive squares in a Galois field $F_q$ are given.

Let $l\ge 2$ be a positive integer, ${𝔽}_q$ a finite field of cardinality $q$ with $q\equiv 1(modl)$. In this paper, inspired by [6, 3, 4] and using a slightly different method, we study the fluctuations in the number of ${𝔽}_q$-points on the curve ${ℂ}_F$ given by the affine model ${ℂ}_F:Y^l=F\left(X\right)$, where $F$ is drawn at random uniformly from the set of all monic $l$-th power-free polynomials $F\in {𝔽}_q\left[X\right]$ of degree $d$ as $d\to \infty$. The method also enables us to study the fluctuations in the number of ${𝔽}_q$-points on the same family of curves arising from the set of monic irreducible...

Let $K$ be a finite field and $Q\in K\left[T\right]$ a polynomial of positive degree. A function $f$ on $K\left[T\right]$ is called (completely) $Q$-additive if $f(A+BQ)=f\left(A\right)+f\left(B\right)$, where $A,B\in K\left[T\right]$ and $deg\left(A\right)\<deg\left(Q\right)$. We prove that the values $(f_1\left(A\right),...,f_d\left(A\right))$ are asymptotically equidistributed on the (finite) image set $\left\{\right(f_1\left(A\right),...,$$f_d...$