Related papers: Number of $k$-normal elements over a finite field

We give a necessary and sufficient condition for a type of generalized power series to be algebraic over the ring of power series with coefficients in a finite field. This result extend a classical theorem of Huang-Stefanescu.

Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$ represents a finite extension of degree $n$ of the finite field ${\mathbb{F}_{q}}$. In this article, we investigate the existence of $m$ elements in arithmetic…

This paper gives elements in the (skew) center of the generalized quantum group corresponding to its irreducible finite dimensional modules. Finally we give a conjecture stating that those must form a basis of the center.

Let $F(z)$ be a $k$-regular series in $\mathbb{Z}[[z]]$ and $b$ be an integer with $b\ge2$. Bell, Bugeaud and Coons [BelBC] proved that $F(\frac{1}{b})$ is either rational or transcendental. In [Mi], we introduce a generalized $k$-regular…

Let $n$ and $k$ be positive integers, and $f_n(k)$ (resp. $g_n(k)$) be the number of unital subrings (resp. unital irreducible subrings) of $\mathbb{Z}^n$ of index $k$. The numbers $f_n(k)$ are coefficients of certain zeta functions of…

We prove independence of normality to different bases We show that the set of real numbers that are normal to some base is Sigma^0_4 complete in the Borel hierarchy of subsets of real numbers. This was an open problem, initiated by…

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an…

An element of a finitely generated non-Abelian free group F(X) is said to be filling if that element has positive translation length in every very small action of F(X) on an $\mathbb{R}$-tree. We give a proof that the set of filling…

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 G be a semi-simple algebraic group over a finitely generated field K of characteristic zero, and let \Gamma < G(K) be a finitely generated Zariski-dense subgroup. In this note we prove that the set of K-generic elements of \Gamma (whose…

Let $\Bbbk$ be an algebraically closed field of characteristic $p>2$. Let $\mathcal{O}_n=\Bbbk[X_1,\ldots,X_n]/(X_1^p,\ldots, X_n^p)$, a truncated polynomial ring in $n$ variables, and denote by $\mathcal{L}$ the derivation algebra of…

Finite elements, which are well-known and studied in the framework of vector lattices, are investigated in $\ell$-algebras, preferably in $f$-algebras, and in product algebras. The additional structure of an associative multiplication leads…

We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobinski…

As defined by Guralnick and Saxl given a nonabelian simple group $S$ and its nonidentity automorphism $x$, a natural number $\alpha_S(x)$ does not exceed a natural number $m$ if some $m$ conjugates of $x$ in the group $\langle x,S\rangle$…

In combinatorial group testing problems Questioner needs to find a special element $x \in [n]$ by testing subsets of $[n]$. Tapolcai et al. introduced a new model, where each element knows the answer for those queries that contain it and…

Fock and Goncharov conjectured that the indecomposable universally positive (i.e., atomic) elements of a cluster algebra should form a basis for the algebra. This was shown to be false by Lee-Li-Zelevinsky. However, we find that the theta…

Let $o(G)$ be the average order of the elements of $G$, where $G$ is a finite group. We show that there is no polynomial lower bound for $o(G)$ in terms of $o(N)$, where $N\trianglelefteq G$, even when $G$ is a prime-power order group and…

In this paper, we show that the concatenation of the Fibonacci sequence is \textit{normal} in base $10$, meaning every string of a given length, $k$, occurs as frequently as every other string of length $k$ (there are as many $1$'s as $2$'s…

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…

We establish a new asymptotic formula for the number of polynomials of degree $n$ with $k$ prime factors over a finite field $\mathbb{F}_q$. The error term tends to $0$ uniformly in $n$ and in $q$, and $k$ can grow beyond $\log n$.…