Related papers: Permutation binomials over finite fields

Recently, Jiang et al. \cite{JIANG2025102522} obtained several classes of Permutation Polynomial of the form $x+\gamma\operatorname{Tr}_q^{q^2}(h(x))$ over finite fields $\mathbb{F}_{q^2},q=2^n$. In this paper, we find the compositional…

For each integer k >= 2, let F(k) denote the largest n for which there exists a permutation \sigma \in S_n, all of whose patterns of length k are distinct. We prove that F(k) = k + \lfloor \sqrt{2k-3} \rfloor + e_k, where e_k \in {-1,0} for…

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 $\mathbb{F}_q$,…

Let $p\equiv -q \equiv 5\pmod 8$ be two prime integers. In this paper, we investigate the unit groups of the fields $ L_1 =\mathbb{Q}(\sqrt 2, \sqrt{p}, \sqrt{q}, \sqrt{-1} )$ and $ L_1^+=\mathbb{Q}(\sqrt 2, \sqrt{p}, \sqrt{q} )$.…

We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{\sigma(1)} \cdots x_{\sigma(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $\sigma$…

Let $k$ be a positive integer and $S_{2k}={\tt x}+{\tt x}^4+...+{\tt x}^{4^{2k-1}}\in\Bbb F_2[{\tt x}]$. It was recently conjectured that ${\tt x}+S_{2k}^{4^{2k}}+S_{2k}^{4^k+3}$ is a permutation polynomial of $\Bbb F_{4^{3k}}$. In this…

In this note, we prove multiplicity one theorems for generalized modular functions (GMF), in terms of their q-exponents, and make a general statement about the nature of values that the prime q-exponents of a GMF can take. We shall also…

In this note we continue the analysis of permutations that avoid substrings j(j+k), 1 <= j <= n-k, k < n, as well as substrings j(j+k) (mod n), 1 <= j <= n. In the first case the number of such permutations can be obtained from recursions…

Let F* be the field of q elements and let P(n,q) denote the projective space of dimension n-1 over F*. We construct a family H^{n}_{k,i} of combinatorial homology modules associated to P(n,q) for a coefficient field F of positive…

Let $\gamma_n=[x_1,\dots,x_n]$ be the $n$th lower central word. Denote by $X_n$ the set of $\gamma_n$-values in a group $G$ and suppose that there is a number $m$ such that $|g^{X_n}|\leq m$ for each $g\in G$. We prove that…

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…

For any prime number $p$, positive integers $m, k, n$ satisfying ${\rm gcd}(p,n)=1$ and $\lambda_0\in \mathbb{F}_{p^m}^\times$, we prove that any $\lambda_0^{p^k}$-constacyclic code of length $p^kn$ over the finite field $\mathbb{F}_{p^m}$…

We investigate permutation polynomials F over finite fields F_{p^n} whose generalized derivative maps x -> F(x + a) - cF(x) are themselves permutations for all nonzero shifts a. This property, termed perfect c-nonlinearity (PcN), represents…

We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set…

For a fixed quadratic irreducible polynomial $f$ with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes $p$ such that $f(p)$ has at most 4 prime factors, improving a classical result of Richert who…

In this essay, we see how prime cyclotomic fields (cyclotomic fields obtained by adjoining a primitive p-th root of unity to Q, where p is an odd prime) can lead to elegant proofs of number theoretical concepts. We namely develop the notion…

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…

Suppose that $n$ is $0$ or $4$ modulo $6$. We show that there are infinitely many primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime, and obtain an asymptotic for their number. In particular, when $n = 4$ we verify the `Gaussian…

It is shown that two vectors with coordinates in the finite $q$-element field of characteristic $p$ belong to the same orbit under the natural action of the symmetric group if each of the elementary symmetric polynomials of degree…

Very recently, Tu et al. presented a sufficient condition about $(a_1,a_2,a_3)$, see Theorem 1.1, such that $f(x) = x^{3\cdot 2^m} + a_1 x^{2^{m+1}+1}+ a_2 x^{2^m+2} + a_3 x^3$ is a class of permutation polynomials over $\gf_{2^{n}}$ with…