Related papers: M\"obius-Frobenius maps on irreducible polynomials

For all finite fields of $q$ elements where $q\equiv1\pmod4$ we have constructed permutation polynomials which have order 2 as permutations, and have 3 terms, or 4 terms as polynomials. Explicit formulas for their coefficients are given in…

In this paper, we study the sum of additive characters over finite fields, with a focus on those of specified \(\mathbb{F}_q\)-Order. We establish a general formula for these character sums, providing an additive analogue to classical…

We prove that super strongly fractal groups acting on regular rooted trees have null fixed-point proportion. In particular, we show that the fixed-point proportion of an infinite family of iterated monodromy groups of exceptional complex…

In this paper we prove the assertion that the number of monic cubic polynomials $F(x) = x^3 + a_2 x^2 + a_1 x + a_0$ with integer coefficients and irreducible, Galois over $\mathbb{Q}$ satisfying $\max\{|a_2|, |a_1|, |a_0|\} \leq X$ is…

Let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a prime power and, for each integer $n\ge 1$, let $\mathbb F_{q^n}$ be the unique $n$-degree extension of $\mathbb F_q$. The $\mathbb F_q$-orders of an element in…

We prove that if $G$ is a finite simple group, then all irreducible complex representations of $G$ by be realized over the real numbers if and only if every element of $G$ may be written as a product of two involutions in $G$. This follows…

We study the set $\mathcal{L}_{F}$ of all $F$-vector spaces $L(P)$ where $P$ is monic and splits over $F$ and $L(Q)$ denotes the set of linear recurrence sequences over $F$ with characteristic polynomial $Q$. We show that $\mathcal{L}_{F}$…

We propose graph theoretic equivalents for existence of a finite projective plane. We then develop a new approach and see that the problem of existence of a finite projective plane of order n is linked up with a subset of sharply 2…

We obtain estimates on the number $|\mathcal{A}_{\boldsymbol{\lambda}}|$ of elements on a linear family $\mathcal{A}$ of monic polynomials of $\mathbb{F}_q[T]$ of degree $n$ having factorization pattern…

Let $\mathbb F_q$ be a finite field and $n$ a positive integer. In this article, we prove that, under some conditions on $q$ and $n$, the polynomial $x^n-1$ can be split into irreducible binomials $x^t-a$ and an explicit factorization into…

Planar functions are mappings from a finite field $\mathbb{F}_q$ to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between…

The main result of this paper is the following theorem. Let q be a prime, A an elementary abelian group of order q^3. Suppose that A acts as a coprime group of automorphisms on a profinite group G in such a manner that C_G(a)' is periodic…

We introduce computable actions of computable groups and prove the following versions of effective Birkhoff's ergodic theorem. Let $\Gamma$ be a computable amenable group, then there always exists a canonically computable tempered two-sided…

We examine the question of when a quadratic polynomial f(x) defined over a number field K can have a newly reducible nth iterate, that is, f^n(x) irreducible over K but f^{n+1}(x) reducible over K, where f^n denotes the nth iterate of f.…

The aim of this paper is to study the $p$-Frobenius vector of affine semigroups $S\subset \mathbb N^q$; that is, the maximum element, with respect to a graded monomial order, with at most $p$ factorizations in $S$. We produce several…

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…

Consider a finite l-group acting on the affine space of dimension n over a field k, whose characteristic differs from l. We prove the existence of a fixed point, rational over k, in the following cases: --- The field k is p-special for some…

Every nonabelian finite simple group of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, has a presentation with a bounded number of generators and relations and total length $O(\log n…

Let $\bfG$ be a connected reductive algebraic group defined over $\F_q$, where $q$ is a power of a prime $p$ that is good for $\bfG$. Let $F$ be the Frobenius morphism associated with the $\FF_q$-structure on $\bfG$ and set $G = \bfG^F$,…

We construct families of rational functions $f \colon \bP^1_k \to \bP^1_k$ of degree $d \geq 2$ over a perfect field $k$ whose associated fixed-point processes fail to be martingales. Conversely, for any normal variety $X \subset…