Related papers: Low-degree permutation rational functions over fin…

Let $p$ be a prime and $n$ be a positive integer. Let $f_b(X)=X+(X^p-X+b)^{-1}$, where $b\in\Bbb F_{p^n}$ is such that $\text{Tr}_{p^n/p}(b)\ne 0$. In 2008, Yuan et al. \cite{Yuan-Ding-Wang-Pieprzyk-FFA-2008} showed that for $p=2,3$, $f_b$…

Let F_{q^n} be the field of order q^n, and let Tr be the trace map from F_{q^n} to its q-element subfield. We exhibit nine sequences of polynomials of the form f(x):=x+c*Tr(x^k), with c in F_{q^n}, such that for each polynomial the function…

We study the number of real rational degree n functions (considered up to linear fractional transformations of the independent variable) with a given set of 2n-2 distinct real critical values. We present a combinatorial reformulation of…

We find all polynomials f,g,h over a field K such that g and h are linear and f(g(x))=h(f(x)). We also solve the same problem for rational functions f,g,h, in case the field K is algebraically closed.

In solving $q$-difference equations, and in the definition of $q$-special functions, we encounter formal power series in which the $n$th coefficient is of size $q^{-\binom{n}{2}}$ with $q\in(0,1)$ fixed. To make sense of these formal…

Let $\mathbb F_q$ be the finite field with $q$ elements, $f, g\in \mathbb F_q[x]$ be polynomials of degree at least one. This paper deals with the asymptotic growth of certain arithmetic functions associated to the factorization of the…

Let $\mathbb{F}_q$ denote the finite fields with $q$ elements. The permutation behavior of several classes of infinite families of permutation polynomials over finite fields have been studied in recent years. In this paper, we continue with…

Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree $8$ over $\mathbb{F}_{2^r}$ with $r>3$. By [J. Number Theory 176 (2017) 466-66], a polynomial $f$ of degree $8$ over $\mathbb{F}_{2^r}$ is…

For a positive integer $N$ divisible by $4$, let $\mathcal{O}^1_N(\mathbb{Q})$ be the ring of weakly holomorphic modular functions for the congruence subgroup $\Gamma^1(N)$ with rational Fourier coefficients. We present explicit generators…

We determine all permutations in two large classes of polynomials over finite fields, where the construction of the polynomials in each class involves the denominators of a class of rational functions generalizing the classical Redei…

The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions. As a result of this, new characterization theorems for homomorphisms and derivations can be given. More exactly, in this paper the…

We establish a sub-convexity estimate for Rankin-Selberg $L$-functions in the combined level aspect, using the circle method. If $p$ and $q$ are distinct prime numbers, $f$ and $g$ are non-exceptional newforms (modular or Maass) for the…

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. In all the examples emerging from physics, the minimal linear differential operators annihilating these diagonals…

Using dynamical methods we give a new proof of the theorem saying that if $A,B,X$ are rational functions of degree at least two such that $A\circ X=X\circ B$ and $\mathbb C(B,X)=\mathbb C(z)$, then the Galois closure of the field extension…

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a…

We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain $L$-functions of degree two and higher in $\mathbb{F}_q[t]$, in the limit as $q\to\infty$. This is achieved by establishing…

Questions concerning small fractional parts of polynomials and pseudo-polynomials have a long history in analytic number theory. In this paper, we improve on earlier work by Madritsch and Tichy. In particular, let $f=P+\phi$ where $P$ is a…

There exist homogeneous polynomials $f$ with $\mathbb Q$-coefficients that are sums of squares over $\mathbb R$ but not over $\mathbb Q$. The only systematic construction of such polynomials that is known so far uses as its key ingredient…

In their 2015 paper, Mertens and Rolen prove that for a certain level 6 "almost holomorphic" modular function $P$, the degree of $P(\tau)$ over $\mathbb{Q}$ for quadratic $\tau$ is as large as expected, settling a conjecture of Bruinier and…

We determine all F,G in C[X] of degree at least 2 for which the semigroup generated by F and G under composition is not the free semigroup on the letters F and G. We also solve the same problem for F,G in X^2 C[[X]], and prove partial…