arXiv subjects

Andrea Ferraguti

Publications and source records attributed to Andrea Ferraguti.

21 records · Page 2Linked to original sources

arXiv2016

On sets of irreducible polynomials closed by composition

Let $\mathcal S$ be a set of monic degree $2$ polynomials over a finite field and let $C$ be the compositional semigroup generated by $\mathcal S$. In this paper we establish a necessary and sufficient condition for $C$ to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set $\mathcal S$. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree $2$ polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions.

math.NT↗

arXiv2016

On the existence of infinite, non-trivial $F$-sets

In this paper we prove a conjecture of J. Andrade, S. J. Miller, K. Pratt and M. Trinh, showing the existence of a non trivial infinite $F$-set over $\mathbb F_q[x]$ for every fixed $q$. We also provide the proof of a refinement of the conjecture, involving the notion of width of an $F$-set, which is a natural number encoding the complexity of the set.

math.NT↗

arXiv2015

On Mertens-Cesàro Theorem for Number Fields

Let $K$ be a number field with ring of integers $\mathcal O$. After introducing a suitable notion of density for subsets of $\mathcal O$, generalizing that of natural density for subsets of $\mathbb Z$, we show that the density of the set of coprime $m$-tuples of algebraic integers is ${1/ζ_K(m)}$, where $ζ_K$ is the Dedekind zeta function of $K$.

math.NT↗