Related papers: Permutation Polynomials with Carlitz Rank 2

This paper provides an algorithmic generalization of Dickson's method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson's idea is to formulate from Hermite's criterion several polynomial…

We present a general technique for obtaining permutation polynomials over a finite field from permutations of a subfield. By applying this technique to the simplest classes of permutation polynomials on the subfield, we obtain several new…

Recently, Tu, Zeng, Li, and Helleseth considered trinomials of the form $f(X)=X+aX^{q(q-1)+1}+bX^{2(q-1)+1}\in\Bbb F_{q^2}[X]$, where $q$ is even and $a,b\in\Bbb F_{q^2}^*$. They found sufficient conditions on $a,b$ for $f$ to be a…

Let $q$ be a prime power. We determine all permutation trinomials of $\Bbb F_{q^2}$ of the form $ax+bx^q+x^{2q-1}\in\Bbb F_{q^2}[x]$. The subclass of such permutation trinomials of $\Bbb F_{q^2}$ with $a,b\in\Bbb F_q$ was determined in a…

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…

In this paper, we present three classes of complete permutation monomials over finite fields of odd characteristic. Meanwhile, the compositional inverses of these complete permutation polynomials are also proposed.

We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[x] of f such that h has at most t…

Permutation polynomials over finite fields are an interesting subject due to their important applications in the areas of mathematics and engineering. In this paper we investigate the trinomial $f(x)=x^{(p-1)q+1}+x^{pq}-x^{q+(p-1)}$ over…

Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, communication…

Let $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to construct…

We construct classes of permutation polynomials over F_{Q^2} by exhibiting classes of low-degree rational functions over F_{Q^2} which induce bijections on the set of (Q+1)-th roots of unity in F_{Q^2}. As a consequence, we prove two…

A family $f_t(z)$ of polynomials over a number field $K$ will be called \emph{weighted homogeneous} if and only if $f_t(z)=F(z^e, t)$ for some binary homogeneous form $F(X, Y)$ and some integer $e\geq 2$. For example, the family $z^d+t$ is…

We study permutation polynomials through the device of the \emph{polypermutation group} of an associative ring $R$, denoted by ${\rm Pgr}(R)$. We derive some basic properties and compute the cardinality of ${\rm Pgr}(\mathbb{Z}/p^k)$ when…

A permutation $\pi$ contains a permutation $\sigma$ as a pattern if it contains a subsequence of length $|\sigma|$ whose elements are in the same relative order as in the permutation $\sigma$. This notion plays a major role in enumerative…

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over $\mathbb{F}_q[t]$ satisfying a certain divisibility condition…

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 consider the problem of counting the number of linear transformation shift registers (TSRs) of a given order over a finite field. We derive explicit formulae for the number of irreducible TSRs of order two. An interesting connection…

In this paper, we propose linear maps over the space of all polynomials $f(x)$ in $\mathbb{F}_q[x]$ that map $0$ to itself, through their evaluation map. Properties of these linear maps throw up interesting connections with permutation…

The (strong) Bruhat order for permutations provides a partial ordering defined as follows: two permutations are comparable if one can be obtained from the other by a sequence of adjacent transpositions that each increase the number of…

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a prime power and $n$ be a positive integer. In this paper, we explore the factorization of $f(x^{n})$ over $\mathbb{F}_q$, where $f(x)$ is an irreducible polynomial…