Related papers: A proof of a conjecture on permutation trinomials

In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over $\mathbb{F}_3^{2k}$. In addition, new examples and generalizations of some families of permutation polynomials of $\mathbb{F}_{3^k}$ and…

Let $p>3$ and consider a prime power $q=p^h$. We completely characterize permutation polynomials of $\mathbb{F}_{q^2}$ of the type $f_{a,b}(X) = X(1 + aX^{q(q-1)} + bX^{2(q-1)}) \in \mathbb{F}_{q^2}[X]$. In particular, using connections…

We prove a conjecture by Tu, Zeng, Li, and Helleseth concerning trinomials $f_{\alpha,\beta}(x)= x + \alpha x^{q(q-1)+1} + \beta x^{2(q-1)+1} \in \mathbb{F}_{q^2}[x]$, $\alpha\beta \neq 0$, $q$ even, characterizing all the pairs…

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.

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are…

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…

We consider four classes of polynomials over the fields $\mathbb{F}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+Ax^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+Ax^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+Ax^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+Ax^{q}-Bx$,…

Let $\mathbb F_q$ denote the finite field with $q$ elements. In this paper we use the relationship between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements $a\in \mathbb F_q$ for…

Let $q>2$ be a prime power and $f={\tt x}^{q-2}+t{\tt x}^{q^2-q-1}$, where $t\in\Bbb F_q^*$. It was recently conjectured that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following holds: (i) $t=1$, $q\equiv…

Let $f=a{\tt x} +b{\tt x}^q+{\tt x}^{2q-1}\in\Bbb F_q[{\tt x}]$. We find explicit conditions on $a$ and $b$ that are necessary and sufficient for $f$ to be a permutation polynomial of $\Bbb F_{q^2}$. This result allows us to solve a related…

Let $f=a\x+\x^{3q-2}\in\Bbb F_{q^2}[\x]$, where $a\in\Bbb F_{q^2}^*$. We prove that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following occurs: (i) $q=2^e$, $e$ odd, and $a^{\frac{q+1}3}$ is a primitive…

Let $\Bbb F_q$ be the finite field with $q$ elements and let $p=\text{char}\,\Bbb F_q$. It was conjectured that for integers $e\ge 2$ and $1\le a\le pe-2$, the polynomial $X^{q-2}+X^{q^2-2}+\cdots+X^{q^a-2}$ is a permutation polynomial of…

Let $f=ax+x^{r(q-1)+1}\in \mathbb{F}_{q^2}^*[x], r\in \{5,7\}.$ We give explicit conditions on the values $(q,a)$ for which $f$ is a permutation polynomials of $\mathbb{F}_{q^2}.$

Permutation polynomials have many applications in finite fields theory, coding theory, cryptography, combinatorial design, communication theory, and so on. Permutation binomials of the form $x^{r}(x^{q-1}+a)$ over $\mathbb{F}_{q^2}$ have…

Permutation polynomials are of particular significance in several areas of applied mathematics, such as Coding theory and Cryptography. Many recent constructions are based on the Akbary-Ghioca-Wang (AGW) criterion. Along this line of…

Let $q>2$ be a prime power and $f=-{\tt x}+t{\tt x}^q+{\tt x}^{2q-1}$, where $t\in\Bbb F_q^*$. We prove that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following occurs: (i) $q$ is even and…

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…

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…

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…

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation…