Related papers: Upper bounds on cyclotomic numbers

We construct explicitly in any finite field of the form Fq[x]/(x^m-a) elements with multiplicative order at least 2^{(2m)^(1/2)}

Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees…

The purpose of this article is to describe a reduction of the slicing problem to the study of the parameter I_1(K,Z_q^o(K))=\int_K ||< :, x> ||_{L_q(K)}dx. We show that an upper bound of the form I_1(K,Z_q^o(K))\leq C_1q^s\sqrt{n}L_K^2,…

We find a new lower bound for the maximal number of zeros to harmonic polynomials, $p(z)+\overline{q(z)}$, when ${\rm deg}\, p = n$ and ${\rm deg}\, q = n-2$.

We provide numerical bounds on the Crouzeix ratiofor KLS matrices $A$ which have a line segment on the boundary of the numerical range. The Crouzeix ratio is the supremum over all polynomials $p$ of the spectral norm of $p(A)$ dividedby the…

We prove a new upper bound for the number of smooth values of a polynomial with integer coefficients. This improves Timofeev's previous result unless the polynomial is a product of linear polynomials with integer coefficients. As an…

For each odd prime $p$, let $\zeta_p$ denote a primitive $p$-th root of unity. In this paper, we study the determinants of some matrices with cyclotomic unit entries. For instance, we show that when $p\equiv 3\pmod4$ and $p>3$ the…

We give an upper bound for the number elliptic Carmichael numbers $n \le x$ that have recently been introduced by J. H. Silverman. We also discuss several possible ways for further improvements.

A subset of $\mathbb{F}_q^2$ is called an arc if it does not contain three collinear points. We show that there are at most $\binom{(1 + o(1))q}{m}$ arcs of size $m \gg q^{1/2} (\log q)^{3/2}$, nearly matching a trivial lower bound…

It is known that two consecutive coefficients of a ternary cyclotomic polynomial $\Phi_{pqr}(x)=\sum_k a_{pqr}(k)x^k$ differ by at most one. In this paper we give a criterion on $k$ to satisfy $|a_{pqr}(k)-a_{pqr}(k-1)|=1$. We use this to…

Recently, Agievich proposed an interesting upper bound on binomial coefficients in the de Moivre-Laplace form. In this article, we show that the latter bound, in the specific case of a central binomial coefficient, is larger than the one…

Let $\Phi_{n}(q)$ denote the $n$-th cyclotomic polynomial in $q$. Recently, Guo and Schlosser [Constr. Approx. 53 (2021), 155--200] put forward the following conjecture: for an odd integer $n>1$, \begin{align*}…

We consider perturbed pendulum-like equations on the cylinder of the form $ \ddot x+\sin(x)= \varepsilon \sum_{s=0}^{m}{Q_{n,s} (x)\, \dot x^{s}}$ where $Q_{n,s}$ are trigonometric polynomials of degree $n$, and study the number of limit…

We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree $2q$, where $q$ is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units…

We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{\'e}lyi, and Littmann~\cite{BE1}…

We derive several existence results concerning cycle types and, more generally, the "mapping behavior" of complete mappings. Our focus is on so-called first-order cyclotomic mappings, which are functions on a finite field $\mathbb{F}_q$…

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 $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to construct…

This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with…

We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for…