Related papers: $\mathbb{F}_q$-primitive points on varieties over …

Let $f=a_0+ a_{1}x+\cdots+a_m x^m\in \Bbb{Z}[x]$ be a primitive polynomial. Suppose that there exists a positive real number $\alpha$ such that $|a_m| \alpha^m>|a_0|+|a_1|\alpha+\cdots+|a_{m-1}|\alpha^{m-1}$. We prove that if there exist…

Let $\Delta_x f(x,y)=f(x+1,y)-f(x,y)$ and $\Delta_y f(x,y)=f(x,y+1)-f(x,y)$ be the difference operators with respect to $x$ and $y$. A rational function $f(x,y)$ is called summable if there exist rational functions $g(x,y)$ and $h(x,y)$…

Properties of the $q$-ultraspherical polynomials for $q$ being a primitive root of unity are derived using a formalism of the $so_q(3)$ algebra. The orthogonality condition for these polynomials provides a new class of trigonometric…

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…

Let $F=x^2+y^2-z^2$, $x_0 \in \mathbb{Z}^3$ primitive with $F(x_0)=0$, and $\Gamma \leq SO_F(\mathbb{Z})$ be a finitely generated thin subgroup. We consider the resulting thin orbits of Pythagorean triples $x_0 \cdot \Gamma$ - specifically…

We investigate finite sets of rational functions $\{ f_{1},f_{2}, \dots, f_{r} \}$ defined over some number field $K$ satisfying that any $t_{0} \in K$ is a $K_{p}$-value of one of the functions $f_{i}$ for almost all primes $p$ of $K$. We…

Let $p$ be a prime and $q$ a power of $p$. For $n\ge 0$, let $g_{n,q}\in\Bbb F_p[{\tt x}]$ be the polynomial defined by the functional equation $\sum_{a\in\Bbb F_q}({\tt x}+a)^n=g_{n,q}({\tt x}^q-{\tt x})$. When is $g_{n,q}$ a permutation…

We prove that for all $q>61$, every non-zero element in the finite field $\mathbb{F}_{q}$ can be written as a linear combination of two primitive roots of $\mathbb{F}_{q}$. This resolves a conjecture posed by Cohen and Mullen.

Let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a power of a prime. We discuss recursive methods for constructing irreducible polynomials over $\mathbb F_q$ of high degree using rational transformations. In particular,…

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation…

Let $S$ be a smooth cubic surface over a finite field $\mathbb F_q$. It is known that $\#S(\mathbb F_q) = 1 + aq + q^2$ for some $a \in \{-2,-1,0,1,2,3,4,5,7\}$. Serre has asked which values of a can arise for a given $q$. Building on…

All binary plane curves of degree less than 7 are examined for curves with a large number of Fq rational points on their smooth model, for q = 2^m ; m = 3, 4,...,11. Previous results are improved, and many new curves are found meeting or…

A nonempty subset A of {1,2,...,n} is called primitive if gcd(A)=1. Let f(n) and f_k(n) denote, respectively, the number of primitive subsets and the number of primitive subsets of cardinality k of {1,2,...,n}. Recursion formulas and…

The bivariate series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ %(where $(q,x)\in {\bf C}^2$, $|q|<1$) defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $\theta (q,.)$ is an entire function. For $q\in (-1,0)$ the…

We show that univariate trinomials $x^n + ax^s + b \in \mathbb{F}_q[x]$ can have at most $\delta \Big\lfloor \frac{1}{2} +\sqrt{\frac{q-1}{\delta}} \Big\rfloor$ distinct roots in $\mathbb{F}_q$, where $\delta = \gcd(n, s, q - 1)$. We also…

A primitive prime divisor of an element a_n of a sequence (a_1,a_2,a_3,...) is a prime P that divides a_n, but does not divide a_m for all m < n. The Zsigmondy set Z of the sequence is the set of n such that a_n has no primitive prime…

Let $\alpha$ and $\beta$ be irrational real numbers and $0<\F<1/30$. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy $\|q\alpha\|\cdot\|q\beta\|<\F$. If we choose $\F$ as a function of $Q$ we get…

We study the number of points in the family of plane curves defined by a trinomial \[ \mathcal{C}(\alpha,\beta)= \{(x,y)\in\mathbb{F}_q^2\,:\,\alpha x^{a_{11}}y^{a_{12}}+\beta x^{a_{21}}y^{a_{22}}=x^{a_{31}}y^{a_{32}}\} \] with fixed…

Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form $y^q-y=f(x)$ with $f\in\Fqr[x]$, on which the…

We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their…