相关论文: Third Power of the Reversed Dickson Polynomial ove…

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 $K_{q^n}(a)$ be a Kloosterman sum over the finite field $\F_{q^n}$ of characteristic $p$. In this note so called subfield conjecture is proved in case $p>3$: if $a\ne0$ belongs to the proper subfield $\F_q$ of $\F_{q^n}$, then…

Let $\Lambda(1,n)$ be the $(1,n)$-th Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke-Maass cusp form and $d_3(n)$ denotes the triple divisor function. This paper establishes non-trivial bounds for the averages of these arithmetic functions…

In this note, we compute the probability that a $k\times n$ matrix can be extended to an $n\times n$ invertible matrix over $\F_q[x]$, which turns out to be $(1-q^{k-n})(1-q^{k-1-n})...(1-q^{1-n})$. Connections with Dirichlet's density…

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 are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a…

A formula discovered by L. Carlitz in 1935 finds an interesting application in permutation rational functions of finite fields. It allows us to determine all rational functions of degree three that permute the projective line $\Bbb P^1(\Bbb…

Let $M_{n,r}=(\sum_{i=1}^{n}q_ix_i^r)^{\frac {1}{r}}, r \neq 0$ and $M_{n,0}=\lim_{r \rightarrow 0}M_{n,r}$ be the weighted power means of $n$ non-negative numbers $x_i$ with $q_i > 0$ satisfying $\sum^n_{i=1}q_i=1$. In particular,…

Let $\mathscr{P}_\mathbb{Q}=\{ \alpha^n \; : \; \alpha \in \mathbb{Q}, \; n \ge 2\}$ be the set of rational perfect powers, and let $S \subseteq \mathscr{P}_\mathbb{Q}$ be a finite subset. We prove the existence of a polynomial $f_S \in…

Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is large odd prime power. In this paper, we improve some recent results on the additive energy of the distance set, and on sumsets of the distance set due to Shparlinski (2016).…

We consider the set of monic irreducible polynomials $P$ over a finite field $\mathbb{F}_q$ such that the multiplicative order modulo $P$ of some a in $\mathbb{F}_q(T)$ is divisible by a fixed positive integer $d$. Call $R_q(a,d)$ this set.…

This paper investigates the dynamical properties of Dickson polynomials over finite fields, focusing on the periodicity and structural behavior of their iterated sequences. We introduce and analyze the sequence $[D_n(x, \alpha) \mod (x^q -…

Let F_{q^n} be the field of order q^n, and let Tr be the trace map from F_{q^n} to its q-element subfield. We exhibit nine sequences of polynomials of the form f(x):=x+c*Tr(x^k), with c in F_{q^n}, such that for each polynomial the function…

We introduce the subsum polynomial of a partition $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_k)$ defined by $\mathrm{sp}(\lambda, x)=\prod_{i=1}^k(1+x^{\lambda_i})$. We study the sum of reciprocals of $\mathrm{sp}(\lambda, x)$ over all…

In this paper we compute the sum of the $k$-th powers over any finite commutative unital rings, thus generalizing known results for finite fields, the rings of integers modulo $n$ or the ring of Gaussian integers modulo $n$. As an…

Given a natural number $n \geq 2$, an integer $k$ and for a judiciously chosen $l = l(n)$ we give necessary and sufficient conditions for the polynomial $f_{n,k} = \big( \sum_{i=1}^{l} x_{i}^{n} \big) - k$ to have roots modulo every…

R. Gupta, P. Gahlyan and R.K. Sharma presented three classes of permutation trinomials over $\mathbb{F}_{q^3}$ in Finite Fields and Their Applications. In this paper, we employ the local method to prove that those polynomials are indeed…

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over the distinct residues can sometimes be computed independent of the prime $p$; for example,…

This paper is devoted to the study of $$ U_t(a,q):=\sum_{1\leq n_1<n_2<\cdots<n_t}\frac{q^{n_1+n_2+\cdots+n_t}}{(1+aq^{n_1}+q^{2n_1})(1+aq^{n_2}+q^{2n_2})\cdots(1+aq^{n_t}+q^{2n_t})} $$ when $a$ is one of $0, \pm 1, \pm2$. The idea builds…

We determine the roots in F_{q^3} of the polynomial X^{2q^k+1} + X + c for each positive integer k and each c in F_q, where q is a power of 2. We introduce a new approach for this type of question, and we obtain results which are more…