Related papers: On a Class of Permutation Polynomials and Their In…

In this work, we completely characterize (i) permutation binomials of the form $x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb{F}_{2^n}[x], n = 2^st, a \in \mathbb{F}_{2^{2t}}^{*}$, and (ii) permutation trinomials of the form…

For all finite fields of $q$ elements where $q\equiv1\pmod4$ we have constructed permutation polynomials which have order 2 as permutations, and have 3 terms, or 4 terms as polynomials. Explicit formulas for their coefficients are given in…

Let $p$ be an odd prime and $e$ be a positive integer. We completely explain the permutation binomials and trinomials arising from the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ over $\mathbb{F}_{p^e}$ when…

Let $p$ be a prime and $q=p^e$. We discuss the properties of the reversed Dickson polynomial $D_{n,2}(1,x)$ of the third kind. We also give several necessary conditions for the reversed Dickson polynomial of the third kind $D_{n,2}(1,x)$ to…

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over $\mathbb{F}_{2^m}$ in Zieve's paper. We prove…

For given linear action of a finite group on a lattice and a positive integer q, we prove that the mod q permutation representation is a quasi-polynomial in q. Additionally, we establish several results that can be considered as mod…

Necessary and sufficient conditions on $A,B\in \mathbb{F}_{q^3}^*$ for $f(X)=X^{q^2-q+1}+AX^{q^2}+BX$ being a permutation polynomial of $\mathbb{F}_{q^3}$ are investigated via a connection with algebraic varieties over finite fields.

Up to linear transformations, we give a classification of all permutation polynomials of degree $7$ over $\mathbb{F}_{q}$ for any odd prime power $q$, with the help of the SageMath software.

Permutation polynomials over finite fields are an interesting and constantly active research subject of study for many years. They have important applications in areas of mathematics and engineering. In recent years, permutation binomials…

In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding $f(X)=X^{q(p-1)+1} +\alpha X^{pq}+X^{q+p-1}$, on finite fields $\mathbb{F}_{q^2}, q=p^k$,…

The ring of dual numbers over a ring $R$ is $R[\alpha] = R[x]/(x^2)$, where $\alpha$ denotes $x+(x^2)$. For any finite commutative ring $R$, we characterize null polynomials and permutation polynomials on $R[\alpha]$ in terms of the…

We discuss the properties and the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ over finite fields. The results in this paper unify and generalize several recently discovered results on…

Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well…

We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We obtain two explicit formulas for these polynomials: a $q$-integral representation and a combinatorial formula. Our main tool is…

In this paper, a construction of complete permutation polynomials over finite fields of even characteristic proposed by Tu et al. recently is generalized in a recursive manner. Besides, several classes of complete permutation polynomials…

Using arbitrary bases for the finite field $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$, we obtain the generalized M\"obius transformations (GMTs), which are a class of bijections between the projective geometry $PG(n-1,q)$ and the set of roots…

In this paper, we introduce an equivariant version of the characteristic quasi-polynomials as the permutation characters on the complement of mod $q$ hyperplane arrangements. We prove that the permutation character is a quasi-polynomial in…

Let $\mathbb{F}_{q}$ be the finite field of characteristic $p$ containing $q = p^{r}$ elements and $f(x)=ax^{n} + x^{m}$ a binomial with coefficients in this field. If some conditions on the gcd of $n-m$ an $q-1$ are satisfied then this…

In this paper, we make use of the classification results of low-degree permutation rational functions together with their geometric properties to investigate rational functions that induce permutations on the multiplicative subgroup mu_q+1,…

For any given polynomial $f$ over the finite field $\mathbb{F}_q$ with degree at most $q-1$, we associate it with a $q\times q$ matrix $A(f)=(a_{ik})$ consisting of coefficients of its powers $(f(x))^k=\sum_{i=0}^{q-1}a_{ik} x^i$ modulo…