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Simone Ugolini

Publications and source records attributed to Simone Ugolini.

17 recordsLinked to original sources
arXiv2020

On the iterations of the maps $ax^{2^k}+b$ and $(a x^{2^k} + b)^{-1}$ over finite fields of characteristic two

The maps $x \mapsto ax^{2^k}+b$ defined over finite fields of characteristic two can be related to the duplication map over binary supersingular elliptic curves. Relying upon the structure of the group of rational points of such curves we can describe the possible cycle lengths of the maps. Then we extend our investigation to the maps $x \mapsto (ax^{2^k}+b)^{-1}$. We also notice some relations between these latter maps and the polynomials $x^{2^k+1} + x +a$, which have been extensively studied in literature.

math.NT↗
arXiv2019

Graded Lie algebras of maximal class of type $n$

Let $n>1$ be an integer. The algebras of the title, which we abbreviate as algebras of type $n$, are infinite-dimensional graded Lie algebras $L= \bigoplus_{i=1}^{\infty}L_i$, which are generated by an element of degree $1$ and an element of degree $n$, and satisfy $[L_i,L_1]=L_{i+1}$ for $i\ge n$. Algebras of type $2$ were classified by Caranti and Vaughan-Lee in 2000 over any field of odd characteristic. In this paper we lay the foundations for a classification of algebras of arbitrary type $n$, over fields of sufficiently large characteristic relative to $n$. Our main result describes precisely all possibilities for the first constituent length of an algebra of type $n$, which is a numerical invariant closely related to the dimension of its largest metabelian quotient.

math.RA↗
arXiv2016

On permutation numerical semigroups

In this paper we introduce the notion of $n$-permutation numerical semigroup. While there are just three $2$-permutation numerical semigroups, there are infinitely many $n$-permutation numerical semigroups if $n > 2$. We construct $16$ families of $3$-permutation numerical semigroups and one family of $n$-permutation numerical semigroups. Finally we present some experimental data, which seem to support a conjecture about the classification of $3$-permutation numerical semigroups.

math.NT↗
arXiv2015

Sequences of irreducible polynomials over odd prime fields via elliptic curve endomorphisms, II

In this paper we extend a previous investigation by us regarding an iterative construction of irreducible polynomials over finite fields of odd characteristic. In particular, we show how it is possible to iteratively construct irreducible polynomials by means of two families of transforms, which we call the $Q_k$ and $\hat{Q}_k$-transforms, related to certain degree two isogenies over elliptic curves, which split the multiplication-by-$2$ map.

math.DS↗
arXiv2013

Sequences of irreducible polynomials without prescribed coefficients over finite fields of even characteristic

In this paper we deal with the construction of sequences of irreducible polynomials with coefficients in finite fields of even characteristic. We rely upon a transformation used by Kyuregyan in 2002, which generalizes the $Q$-transform employed previously by Varshamov and Meyn for the synthesis of irreducible polynomials. While in the iterative procedure described by Kyuregyan the coefficients of the initial polynomial of the sequence have to satisfy certain hypotheses, in the present paper we construct infinite sequences of irreducible polynomials of non-decreasing degree starting from any irreducible polynomial.

math.DS↗
arXiv2013

A graph-based approach to repeating decimals

In this paper we deal with a classical problem in elementary number theory, namely repeating decimals. We show how the digits of the period of the decimal representation of any fraction $\frac{k}{m}$, where $k$ and $m$ are positive integers arbitrarily chosen, can be obtained relying upon the graphs associated with the iteration of a certain map over the finite set $\{0, 1, \dots, 10n-2 \}$ for a suitable integer $n$, which depends on $m$. In the last section of the paper we generalize the results to any arbitrary choice of the base $B \geq 2$ for the representation of the fraction $\frac{k}{m}$.

math.NT↗
arXiv2012

Sequences of binary irreducible polynomials

In this paper we construct an infinite sequence of binary irreducible polynomials starting from any irreducible polynomial $f_0 \in \F_2 [x]$. If $f_0$ is of degree $n = 2^l \cdot m$, where $m$ is odd and $l$ is a non-negative integer, after an initial finite sequence of polynomials $f_0, f_1, ..., f_{s}$ with $s \leq l+3$, the degree of $f_{i+1}$ is twice the degree of $f_i$ for any $i \geq s$.

math.DS↗
arXiv2012

Sequences of irreducible polynomials without prescribed coefficients over odd prime fields

In this paper we construct infinite sequences of monic irreducible polynomials with coefficients in odd prime fields by means of a transformation introduced by Cohen in 1992. We make no assumptions on the coefficients of the first polynomial $f_0$ of the sequence, which belongs to $\F_p [x]$, for some odd prime $p$, and has positive degree $n$. If $p^{2n}-1 = 2^{e_1} \cdot m$ for some odd integer $m$ and non-negative integer $e_1$, then, after an initial segment $f_0, ..., f_s$ with $s \leq e_1$, the degree of the polynomial $f_{i+1}$ is twice the degree of $f_i$ for any $i \geq s$.

math.NT↗
arXiv2011

Graphs associated with the map $x \mapsto x+x^{-1}$ in finite fields of characteristic two

In this paper we study the structure of the graphs associated with the iterations of the map $x \mapsto x+x^{-1}$ over finite fields of characteristic two. Formulas are given for the length of the cycles and the depth of the trees relying upon the structure of the group of the rational points of Koblitz curves and the congruences of Kloosterman sums modulo powers of 2.

math.NT↗