Let ω be a sequence of positive integers. Given a positive integer n, we define rₙ(ω) = |(a,b) ∈ ℕ × ℕ : a,b ∈ ω, a+b = n, 0 < a < b|. S. Sidon conjectured that there exists a sequence ω such that rₙ(ω) > 0 for all n sufficiently large and, for all ϵ > 0, $li{m}_{n\to \infty }rₙ\left(\omega \right)/n^{ϵ}=0$. P. Erdős proved this conjecture by showing the existence of a sequence ω of positive integers such that log n ≪ rₙ(ω) ≪ log n. In this paper, we prove an analogue of this conjecture in ${}_q\left[T\right]$, where ${}_q$ is a finite field of...

Let $𝒜$ be the algebra of quaternions $ℍ$ or octonions $𝕆$. In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial $f\left(t\right)\in 𝒜\left[t\right]$ has a root in $𝒜$. As a consequence, the Jacobian determinant $\left|J\right(f\left)\right|$ is always non-negative in $𝒜$. Moreover, using the idea of the topological degree we show that a regular polynomial $g\left(t\right)$ over $𝒜$ has also a root in $𝒜$. Finally, utilizing multiplication ($*$) in $𝒜$, we prove various results on the topological degree...

Given a compact set $K\subset {ℂ}^N$, for each positive integer n, let $V^{\left(n\right)}\left(z\right)$ = $V_K^{\left(n\right)}\left(z\right)$ := sup$1/\left(degp\right)V_{p\left(K\right)}\left(p\left(z\right)\right)$: p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions $V_K^{\left(n\right)}$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, $V_K\left(z\right)$:= max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, ${\left|\right|p\left|\right|}_K\le 1$]. Our main result is that if K is regular, then all of the functions $V_K^{\left(n\right)}$ are continuous; and their associated Robin functions ${\varrho }_{V_K^{\left(n\right)}}\left(z\right):=limsu{p}_{\left|\lambda \right|\to \infty }[V_K^{\left(n\right)}\left(\lambda z\right)-log\left(\right|\lambda \left|\right)]$ increase to ${\varrho }_K:={\varrho }_{V_K}$ for all z outside a pluripolar...

Let ${𝔽}_q\left[t\right]$ denote the polynomial ring over ${𝔽}_q$, the finite field of $q$ elements. Suppose the characteristic of ${𝔽}_q$ is not $2$ or $3$. We prove that there exist infinitely many $N\in \mathbb{N}$ such that the set $\{f\in {𝔽}_q\left[t\right]:degf<N\}$ contains a Sidon set which is an additive basis of order $3$.

Let $K=\mathbb{Q}\left(\alpha \right)$ be a pure number field generated by a complex root $\alpha$ of a monic irreducible polynomial $F\left(x\right)=x^{2^r·3^k·7^s}-m\in \mathbb{Z}\left[x\right]$, where $r$, $k$, $s$ are three positive natural integers. The purpose of this paper is to study the monogenity of $K$. Our results are illustrated by some examples.

In this paper, we examine a natural question concerning the divisors of the polynomial $x^n-1$: “How often does $x^n-1$ have a divisor of every degree between $1$ and $n$?” In a previous paper, we considered the situation when $x^n-1$ is factored in $\mathbb{Z}\left[x\right]$. In this paper, we replace $\mathbb{Z}\left[x\right]$ with ${𝔽}_p\left[x\right]$, where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$.

For a finite group $G$ denote by $N\left(G\right)$ the set of conjugacy class sizes of $G$. In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N\left(G\right)=N\left(L\right)$, then $G\cong L$. We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z\left(G\right)=1$ and $N\left(G\right)=N\left(A_i\right)$ is necessarily isomorphic to $A_i$, where $i\in \{2p,2p+1\}$.

We provide a lower bound for the number of distinct zeros of a sum $1+u+v$ for two rational functions $u,v$, in term of the degree of $u,v$, which is sharp whenever $u,v$ have few distinct zeros and poles compared to their degree. This sharpens the “$abcd$-theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface $x^a+y^a+z^c=1$ contains only finitely many rational or elliptic...

Let $C$ be a hyperelliptic curve of genus $g\ge 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\ge 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside...

We study sums and products in a field. Let $F$ be a field with $\mathrm{ch}\left(F\right)\ne 2$, where $\mathrm{ch}\left(F\right)$ is the characteristic of $F$. For any integer $k\ge 4$, we show that any $x\in F$ can be written as $a_1+\cdots +a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1\cdots a_k=1$, and that for any $\alpha \in F\setminus \left\{0\right\}$ we can write every $x\in F$ as $a_1\cdots a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1+\cdots +a_k=\alpha$. We also prove that for any $x\in F$ and $k\in \{2,3,\cdots \}$ there are $a_1,\cdots ,a_{2k}\in F$ such that $a_1+\cdots +a_{2k}=x=a_1\cdots a_{2k}$.

A monomial self-map $f$ on a complex toric variety is said to be $k$-stable if the action induced on the $2k$-cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of $f$, we can find a toric model with at worst quotient singularities where $f$ is $k$-stable. If $f$ is replaced by an iterate one can find a $k$-stable model as soon as the dynamical degrees ${\lambda }_k$ of $f$ satisfy ${\lambda }_k^2\&gt;{\lambda }_{k-1}{\lambda }_{k+1}$. On the other hand, we give examples of monomial maps $f$, where...

We prove that for every ε > 0 and every nonnegative integer w there exist primes $p_1,...,p_w$ such that for $n=p_1...p_w$ the height of the cyclotomic polynomial ${\Phi }_n$ is at least $(1-\epsilon )c_w{M}_n$, where $M_n={\prod }_{i=1}^{w-2}{p}_i^{2^{w-1-i}-1}$ and $c_w$ is a constant depending only on w; furthermore $li{m}_{w\to \infty }{c}_w^{2^{-w}}\approx 0.71$. In our construction we can have $p_i>h(p_1...p_{i-1})$ for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.