Related papers: Counting solutions of special linear equations ove…

Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\dots+x_n^{2^m}=0$ over a finite field of characteristic $p\equiv\pm 3\pmod{8}$. All of the…

Let $\mathbb{F}_q$ stand for the finite field of odd characteristic $p$ with $q$ elements ($q=p^{n},n\in \mathbb{N} $) and $\mathbb{F}_q^*$ denote the set of all the nonzero elements of $\mathbb{F}_{q}$. Let $m$ and $t$ be positive…

Let F_q be the finite field of q elements. Let H be a multiplicative subgroup of F_q^*. For a positive integer k and element b\in F_q, we give a sharp estimate for the number of k-element subsets of H which sum to b.

We classify those finite fields $\mathbb{F}_q$, for $q$ a power of some fixed prime number, whose members are the sum of an $n$-potent element with $n>1$ and a 4-potent element. It is shown that there are precisely ten non-trivial pairs…

Let $N$ denote the number of solutions to the generalized Markoff-Hurwitz-type equation \[(a_1X_1^m+\cdots + a_nX_n^m+a)^k=bX_1\cdots X_n \] over the finite field $\mathbb{F}_q$, where $m,k$ are positive integers, and $a,b,a_i\in…

In this paper we study the set of rational solutions of equations defined by power sums symmetric polynomials with coefficients in a finite field. We do this by means of applying a methodology which relies on the study of the geometry of…

Let $\{ {U_{n}\}_{n \geq 0} }$ be a non-degenerate binary recurrence sequence with positive discriminant. Let $\{p_1,\ldots, p_s\}$ be fixed prime numbers and $\{b_1,\ldots ,b_s\}$ be fixed non-negative integers. In this paper, we obtain…

Let F_q be the finite field with cardinality q, where q is a prime power. Given a finite field extension F_q^n over F_q and a,b in (F_q)^{*}, we investigate in this article the number N_n(a,b) of elements in F_q^n whose norm equals a and…

We consider the linear vector space formed by the elements of the finite fields $\mathbb{F}_q$ with $q=p^r$ over $\mathbb{F}_p$. Let ${a_1,\ldots,a_r}$ be a basis of this space. Then the elements $x$ of $\mathbb{F}_q$ have a unique…

Let $q$ be a prime power and, for each positive integer $n\ge 1$, let $\mathbb F_{q^n}$ be the finite field with $q^n$ elements. Motivated by the well known concept of normal elements over finite fields, Huczynska et al (2013) introduced…

We show explicit estimates on the number of $q$--rational points of an $F_q$--definable affine absolutely irreducible variety of the algebraic closure of the finite field $F_q$ of $q$ elements. Our estimates for a hypersurface significantly…

We prove that if $k$ is a positive integer then for every finite field $\mathbb{F}$ of cardinality $q\neq 2$ and for every positive integer $n$ such that $q^n>(k-1)^4$, every $n\times n$ matrix over $\mathbb{F}$ can be expressed as a sum of…

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

Given $\mathbb{F}_{q^{n}}$, a field with $q^n$ elements, where $q $ is a prime power and $n$ is positive integer. For $r_1,r_2,m_1,m_2 \in \mathbb{N}$, $k_1,k_2 \in \mathbb{N}\cup \{0\}$, a rational function $F = \frac{F_1}{F_2}$ in…

We obtain a bound on the number of solutions of $x^q=x$ in a finite noncommutative algebra over a field with $q$ elements. Furthermore, we completely characterize those rings for which this maximum number is attained.

We study the mean square of sums of the $k$th divisor function $d_k(n)$ over short intervals and arithmetic progressions for the rational function field over a finite field of $q$ elements. In the limit as $q\rightarrow\infty$ we establish…

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).…

Given a finite field $\mathbb F_q$, a positive integer $n$ and an $\mathbb F_q$-affine space $\mathcal A\subseteq \mathbb F_{q^n}$, we provide a new bound on the sum $\sum_{a\in \mathcal A}\chi(a)$, where $\chi$ a multiplicative character…

Let $D$ be a Krull domain admitting a prime element with finite residue field and let $K$ be its quotient field. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $D$,…

We study the set of common F_q-rational zeros of systems of multivariate symmetric polynomials with coefficients in a finite field F_q. We establish certain properties on these polynomials which imply that the corresponding set of zeros…