A quasi-permutation polynomial is a polynomial which is a bijection from one subset of a finite field onto another with the same number of elements. This is a natural generalization of the familiar permutation polynomials. Basic properties of quasi-permutation polynomials are derived. General criteria for a quasi-permutation polynomial extending the well-known Hermite's criterion for permutation polynomials as well as a number of other criteria depending on the permuted domain and range are established....

Introduction. Soit q une puissance d’un nombre premier p et soit ${}_q$ le corps fini à q éléments. Une certaine analogie entre l’arithmétique de l’anneau ℤ des entiers rationnels et celle de l’anneau ${}_q\left[T\right]$ a conduit à étendre à ${}_q\left[T\right]$ de nombreuses questions de l’arithmétique classique. L’équirépartition modulo 1 est une de ces questions. Le corps des nombres réels est alors remplacé par le corps ${}_q\left(\left(T^{-1}\right)\right)$ des séries de Laurent formelles, complété du corps ${}_q\left(T\right)$ des fractions rationnelles pour la valuation à l’infini et...

Let ${𝔽}_q\left[t\right]$ denote the polynomial ring over ${𝔽}_q$, the finite field of $q$ elements. Suppose the characteristic of ${𝔽}_q$ is not $2$ or $3$. We prove that there exist infinitely many $N\in \mathbb{N}$ such that the set $\{f\in {𝔽}_q\left[t\right]:degf<N\}$ contains a Sidon set which is an additive basis of order $3$.

In this article we extend the algebraic theory of polynomial rings, formalized in Mizar [1], based on [2], [3]. After introducing constant and monic polynomials we present the canonical embedding of R into R[X] and deal with both unit and irreducible elements. We also define polynomial GCDs and show that for fields F and irreducible polynomials p the field F[X]/ is isomorphic to the field of polynomials with degree smaller than the one of p.

En utilisant le théorème de Christol, Kamae, Mendès France et Rauzy, nous donnons une démonstration élémentaire de la transcendance de la série formelle $\Pi$ ainsi que d’autres séries formelles à coefficients dans un corps fini.

En 1976, Baum et Sweet ont donné le premier exemple d’une série formelle algébrique de degré $3$ sur ${𝔽}_2\left(T\right)$ ayant un développement en fraction continue dont les quotients partiels sont tous des polynômes en $T$ de degré $1$ ou $2$. Cette série formelle est l’unique solution dans le corps ${𝔽}_2\left(\left(T^{-1}\right)\right)$ de l’équation $T{X}^3+X-T=0$. En 1986, Mills et Robbins ont décrit un algorithme permettant de calculer le développement en fraction continue de la série de Baum et Sweet.Dans cet article, nous considérons les équations plus générales...

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 $deg\left(A\right)\&lt;deg\left(Q\right)$. 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...$