Related papers: Four new classes of permutation trinomials and the…

We investigate a family of permutation polynomials of finite fields of characteristic 2. Through a connection between permutation polynomials and quadratic forms, a general treatment is presented to characterize these permutation…

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over $\mathbb F_{q^2}$, whose numerators are so-called $q$-quadratic…

G.L. Watson \cite{watson1, watson2} introduced a set of transformations, called Watson transformations by most recent authors, in his study of the arithmetic of integral quadratic forms. These transformations change an integral quadratic…

We compute an analogue of Pascal's triangle enriched in bilinear forms over a finite field. This gives an arithmetically meaningful count of the ways to choose $j$ ring homomorphisms into an algebraic closure from an \'etale extension of…

We give necessary and sufficient conditions for a polynomial of the form x^r*(1+x^v+x^(2v)+...+x^(kv))^t to permute the elements of the finite field GF(q). Our results yield especially simple criteria in case (q-1)/gcd(q-1,v) is a small…

For any integer $x$, let $T_x$ denote the triangular number $\frac{x(x+1)}{2}$. In this paper we give a complete characterization of all the triples of positive integers $(\alpha, \beta, \gamma)$ for which the ternary sums $\alpha x^2…

We obtain connection coefficients between $q$-binomial and $q$-trinomial coefficients. Using these, one can transform $q$-binomial identities into a $q$-trinomial identities and back again. To demonstrate the usefulness of this procedure we…

Two classes of ternary bent functions of degree four with two and three terms in the univariate representation that belong to the completed Maiorana-McFarland class are found. Binomials are mappings $\F_{3^{4k}}\mapsto\fthree$ given by…

Let $\mathbb{F}_q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over $\mathbb{F}_{q^3}$ of the family $$ f_{E,A,B,C,D}(X) := EX^2+ AX^{q+1}+ BX^{q^2+1}+CX^{2q} +DX^{2q^2}\in \mathbb{F}_{q}[X]. $$ Using…

An orthomorphism over a finite field $\mathbb{F}_q$ is a permutation $\theta:\mathbb{F}_q\mapsto\mathbb{F}_q$ such that the map $x\mapsto\theta(x)-x$ is also a permutation of $\mathbb{F}_q$. The degree of an orthomorphism of $\mathbb{F}_q$,…

We define a $q$-deformation of the classical ring of integer-valued polynomials which we call the ring of quantum integer-valued polynomials. We show that this ring has a remarkable combinatorial structure and enjoys many positivity…

Let $m, n$ be positive integers such that $m>1$ divides $n$. In this paper, we introduce a special class of piecewise-affine permutations of the finite set $[1, n]:=\{1, \ldots, n\}$ with the property that the reduction $\pmod m$ of $m$…

We provide constructions of bent functions using triples of permutations. This approach is due to Mesnager. In general, involutions have been mostly considered in such a machinery; we provide some other suitable triples of permutations,…

In 1987, Andrews and Baxter introduced six kinds of $q$-trinomial coefficients in exploring the solution of a model in statistical mechanics. In this paper, we give some $q$-supercongruences for the truncated forms of these polynomials.

An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications such as cryptography and coding theory. As far as we know,…

Let $\mu_{q+1}$ denote the set of $(q+1)$-th roots of unity in $\mathbb{F}_{q^2 }$. We construct permutation polynomials over $\mathbb{F}_{q^2}$ by using rational functions of any degree that induce bijections either on $\mu_{q+1}$ or…

We introduce the notion of non commutative truncated polynomial extension of an algebra A. We study two families of these extensions. For the first one we obtain a complete classification and for the second one, which we call upper…

Connected the generalized Goncharov polynomials associated to a pair ($\partial,\mathcal{Z}$) if a delta operator $\partial$ and an interpolation grid $\mathcal{Z}$, introduced by Lorentz, Tringali and Yan in [7], with the theory of…

We construct the new q-extension of Bernoulli numbers and polynomials in this paper. Finally we consider the q-zeta functions which interpolate the new q-extension of Bernoulli numbers and polynomials.

We introduce a new type of reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms $\sigma$ and uses a generalization of the concept of effective order of a difference polynomial.…