Let $\mathbb{F}_q$ be a finite field of odd characteristic. We study R\'edei functions that induce permutations over $\mathbb{P}^1(\mathbb{F}_q)$ whose cycle decomposition contains only cycles of length $1$ and $j$, for an integer $j\geq 2$. When $j$ is $4$ or a prime number, we give necessary and sufficient conditions for a R\'edei permutation of this type to exist over $\mathbb{P}^1(\mathbb{F}_q)$, characterize R\'edei permutations consisting of $1$- and $j$-cycles, and determine their total number. We also present explicit formulas for R\'edei involutions based on the number of fixed points, and procedures to construct R\'edei permutations with a prescribed number of fixed points and $j$-cycles for $j \in \{3,4,5\}$.

Juliane Capaverde Ariane M. Masuda Virgínia M. Rodrigues

For positive integers $u$ and $v$, let $L_u=\begin{bmatrix}1 & 0 \\u&1\end{bmatrix}$ and $R_v=\begin{bmatrix}1 & v \\ 0 & 1\end{bmatrix}$. Let $G_{u,v}$ be the group generated by $L_u$ and $R_v$. In a previous paper, the authors determined a characterization of matrices $M=\begin{bmatrix}a & c \\b&d\end{bmatrix}$ in $G_{u,v}$ when $u,v\geq 3$ in terms of the short continued fraction representation of $b/d$. We extend this result to the case where $u+v> 4$. Additionally, we compute $[\mathscr{G}_{u,v}\colon G_{u,v}]$ for $u,v\geq 1$, extending a result of Chorna, Geller, and Shpilrain.

Sandie Han Ariane M. Masuda Satyanand Singh Johann Thiel

We fix integers $u,v \geq 1$, and consider an infinite binary tree $\mathcal{T}^{(u,v)}(z)$ with a root node whose value is a positive rational number $z$. For every vertex $a/b$, we label the left child as $a/(ua+b)$ and right child as $(a+vb)/b$. The resulting tree is known as the $(u,v)$-Calkin-Wilf tree. As $z$ runs over $[1/u,v]\cap \mathbb{Q}$, the vertex sets of $\mathcal{T}^{(u,v)}(z)$ form a partition of $\mathbb{Q}^+$. When $u=v=1$, the mean row value converges to $3/2$ as the row depth increases. Our goal is to extend this result for any $u,v\geq 1$. We show that, when $z\in [1/u,v]\cap \mathbb{Q}$, the mean row value in $\mathcal{T}^{(u,v)}(z)$ converges to a value close to $v+\log 2/u$ uniformly on $z$.

Sandie Han Ariane M. Masuda Satyanand Singh Johann Thiel

The basic character theory of finite monoids over the complex numbers was developed in the sixties and seventies based on work of Munn, Ponizovsky, McAlister, Rhodes and Zalcstein. In particular, McAlister determined the space of functions spanned by the irreducible characters of a finite monoid over $\mathbb C$ and the ring of virtual characters. In this paper, we present the corresponding results over an arbitrary field. As a consequence, we obtain a quick proof of the theorem of Berstel and Reutenauer that the characteristic function of a regular cyclic language is a virtual character of the free monoid. This is a crucial ingredient in their proof of the rationality of the zeta function of a sofic shift in symbolic dynamics.

Ariane M. Masuda Luciane Quoos Benjamin Steinberg

Let $L_u=\begin{bmatrix}1 & 0\\u & 1\end{bmatrix}$ and $R_v=\begin{bmatrix}1 & v\\0 & 1\end{bmatrix}$ be matrices in $SL_2(\mathbb Z)$ with $u, v\geq 1$. Since the monoid generated by $L_u$ and $R_v$ is free, we can associate a depth to each element based on its product representation. In the cases where $u=v=2$ and $u=v=3$, Bromberg, Shpilrain, and Vdovina determined the depth $n$ matrices containing the maximal entry for each $n\geq 1$. By using ideas from our previous work on $(u,v)$-Calkin-Wilf trees, we extend their results for any $u, v\geq 1$ and in the process we recover the Fibonacci and some Lucas sequences. As a consequence we obtain bounds which guarantee collision resistance on a family of hashing functions based on $L_u$ and $R_v$.

Sandie Han Ariane M. Masuda Satyanand Singh Johann Thiel

Let $\mathbb{F}_q$ be the finite field of order $q$, and $\mathbb P^1(\mathbb{F}_q) = \mathbb F_q\cup \{\infty\}$. Write $(x+\sqrt y)^m$ as $N(x,y)+D(x,y)\sqrt{y}$. For $m\in\mathbb N$ and $a \in \mathbb{F}_q$, the R\'edei function $R_{m,a}\colon \mathbb P^1(\mathbb F_q) \to \mathbb P^1(\mathbb F_q)$ is defined by $N(x,a)/D(x,a)$ if $D(x,a)\neq 0$ and $x\neq\infty$, and $\infty$, otherwise. In this paper we give a complete characterization of all pairs $(m,n)\in\mathbb N^2$ such that the R\'edei permutations $R_{m,a}$ and $R_{n,b}$ have the same cycle structure when $a$ and $b$ have the same quadratic character and $q$ is odd. We explore some relationships between such pairs $(m,n)$, and provide explicit families of R\'edei permutations with the same cycle structure. When a R\'edei permutation has a unique cycle structure that is not shared by any other R\'edei permutation, we call it isolated. We show that the only isolated R\'edei permutations are the isolated R\'edei involutions. Moreover, all our results can be transferred to bijections of the form $mx$ and $x^m$ on certain domains.

Juliane Capaverde Ariane M. Masuda Virgínia M. Rodrigues

In 1999 Khabibullin established the best estimate in Paley problem for a plurisubharmonic function $u$ of finite lower order $0\leq\rho\leq 1$. For $\rho>1$ obtaining a sharp estimate has remained an open question. In this work we solve this problem. We also provide some estimates for the types of the characteristic functions $T(r,u)$ and $M(r,u)$.

Arian Bërdëllima

We compute the universal deformation ring of an odd Galois two dimensional representation of Gal$(M/Q)$ with an upper triangular image, where $M$ is the maximal abelian pro-$p$-extension of $F_{\infty}$ unramified outside a finite set of places S, $F_{\infty}$ being a free pro-$p$-extension of a subextension $F$ of the field $K$ fixed by the kernel of the representation. We establish a link between the latter universal deformation ring and the universal deformation ring of the representation of Gal$(K_S/Q)$, where $K_S$ is the maximal pro-$p$-extension of $K$ unramified outside $S$. We then give some examples. This paper was accepted for publication in the Mathematical Proceedings of the Cambridge philosophical society (May 99).

Ariane Mezard

Given an infinite matrix $M=(m_{nk})$ we study a family of sequence spaces $\ell_M^p$ associated with it. When equipped with a suitable norm $\|\cdot\|_{M,p}$ we prove some basic properties of the Banach spaces of sequences $(\ell_M^p,\|\cdot\|_{M,p})$. In particular we show that such spaces are separable and strictly/uniformly convex for a considerably large class of infinite matrices $M$ for all $p>1$. A special attention is given to the identification of the dual space $(\ell_M^p )^*$. Building on the earlier works of Bennett and J\"agers, we extend and apply some classical factorization results to the sequence spaces $\ell_M^p$.

Arian Bërdëllima Naim L. Braha

Let $X$ be an affine surface admitting a unique affine ruling and a $\mathbb C^*$-action. Assume that the ruling has a unique degenerate fibre and that this fibre is irreducible. In this paper we give a short proof of the following result of Miyanishi and Masuda: the universal covering of $X$ is a hypersurface in the affine 3-space given by the equation $x^my=z^d-1$, where $m>1$.

Hubert Flenner Mikhail Zaidenberg