Related papers: Full Classification of permutation rational functi…

In this note, we prove multiplicity one theorems for generalized modular functions (GMF), in terms of their q-exponents, and make a general statement about the nature of values that the prime q-exponents of a GMF can take. We shall also…

Let $\varphi:\mathbb{P}^1(\mathbb F_q)\to\mathbb{P}^1(\mathbb F_q)$ be a rational map of degree $d>1$ on a fixed finite field. We give asymptotic formulas for the size of image sets $\varphi^n(\mathbb{P}^1(\mathbb F_q))$ as a function of…

Let $q=p^s$ be a prime power, $F$ a field containing a root of unity of order $q$, and $G_F$ its absolute Galois group. We determine a new canonical quotient $\mathrm{Gal}(F_{(3)}/F)$ of $G_F$ which encodes the full mod-$q$ cohomology ring…

This paper concerns the enumeration of isomorphism classes of modules of a polynomial algebra in several variables over a finite field. This is the same as the classification of commuting tuples of matrices over a finite field up to…

In this paper, we study questions of definability and decidability for infinite algebraic extensions ${\bf K}$ of $\mathbb{F}_p(t)$ and their subrings of $\mathcal{S}$-integral functions. We focus on fields ${\bf K}$ satisfying a local…

In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite…

Let $B$ be a fixed rational function of one complex variable of degree at least two. In this paper, we study solutions of the functional equation $A\circ X=X\circ B$ in rational functions $A$ and $X$. Our main result states that, unless $B$…

We complete the proof of the fact that all principal permutation classes generated by a pattern longer than two have a nonrational generating function.

It is well known that constructing codes with good parameters is one of the most important and fundamental problems in coding theory. Though a great many of good codes have been produced, most of them are defined over alphabets of sizes…

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 prove model completeness for the theory of addition and the Frobenius map for certain subrings of rational functions in positive characteristic. More precisely: Let $p$ be a prime number, $\mathbb{F}_{p}$ the prime field with $p$…

Nonsingular plane curves over a finite field $\mathbb{F}_q$ of degree $q+2$ passing through all the $\mathbb{F}_q$-points of the plane admita representation by $3\times 3$ matrices over $\mathbb{F}_q$. We classify their degenerations by…

We provide an explicit bound on the number of periodic points of a rational function defined over a number field, where the bound depends only on the number of primes of bad reduction and the degree of the function, and is linear in the…

This is a sequel to math.AG/0003009. Here we study identities for the Fourier transform of "elementary functions" over finite field containing "exponents" of monomial rational functions. It turns out that these identities are governed by…

Let $r\geq 3$ be a positive integer and $\mathbb{F}_q$ the finite field with $q$ elements. In this paper, we consider the $r$-regular complete permutation property of maps with the form $f=\tau\circ\sigma_M\circ\tau^{-1}$ where $\tau$ is a…

Cohn and Kumar showed the quadratic character of $q$ modulo $p$ gives the sign of the permutation of Hurwitz primes of norm $p$ induced by the Hurwitz primes of norm $q$ under metacommutation. We demonstrate that these permutations are…

In this paper, we present three classes of complete permutation monomials over finite fields of odd characteristic. Meanwhile, the compositional inverses of these complete permutation polynomials are also proposed.

Let $D_n(x;a)$ and $E_n(x;a)\in\mathbb F_q[x]$ be Dickson polynomials of first and second kind respectively, where $\mathbb F_q$ is a finite field with $q$ elements. In this article we show explicitly the irreducible factors these…

We say a power series $\sum_{n=0}^\infty a_n q^n$ is multiplicative if the function $n\mapsto a_n/a_1$ ($n\ge 1$) is so. In this paper, we consider multiplicative power series $f$ such that $f^2$ is also multiplicative. We find various…

If $G$ is a finite group, an irreducible complex-valued character $\chi$ is called rational if $\chi(g)$ is rational for all $g\in G$. Also, a conjugacy class $x^G$ is called rational, if for all irreducible complex-valued character $\chi$,…