arXiv subjects

Giulia Salvatori

Publications and source records attributed to Giulia Salvatori.

4 recordsLinked to original sources

arXiv2026

Transcendence of continued fractions over function fields and a quantitative version of Uchiyama's theorem

Given a field $K$, let $K((T^{-1}))$ be the field of formal power series. Continued fractions in $K((T^{-1}))$ can be defined by analogy with classical real continued fractions and have been widely studied. Some results establish the transcendence of elements of $K((T^{-1}))$ arising from special families of continued fractions, but much remains to be explored. In this paper, assuming that $K$ has characteristic zero, we improve the known analogues of the Maillet--Baker criteria for quasi-periodic continued fractions. A central tool that we prove is a quantitative version of Uchiyama's analogue of Roth's theorem in function fields, which gives an explicit bound for the number of exceptionally good rational approximations to an algebraic power series. This quantitative estimate also yields a Davenport--Roth-type upper bound on the growth of the denominators of the convergents of algebraic elements. Finally, we prove that palindromic continued fractions are either quadratic or transcendental, as in the real case, but using a different proof strategy.

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arXiv2025

The arithmetic of continued fractions in the field of $p$-adic numbers

Continued fractions have been long studied due to their strong properties, such as rational approximation. In this extent, their arithmetic over real numbers has represented an intriguing problem throughout the years. In this paper, we develop the arithmetic of continued fractions over the field of $p$-adic numbers. In particular, we provide a complete methodology to compute the $p$-adic continued fraction of the M\"obius transformation and the bilinear fractional transformation of $p$-adic numbers. These allow any standard arithmetic operation over $p$-adic numbers to be performed. In great contrast with real continued fractions, we prove that the knowledge of arbitrarily many partial quotients of the initial continued fractions is not always sufficient to recover some partial quotients of the transformations. However, we prove that the set of elements for which this is not possible has Haar measure zero in $\mathbb{Q}_p$.

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arXiv2024

Integer Factorization via Continued Fractions and Quadratic Forms

We propose a novel factorization algorithm that leverages the theory underlying the SQUFOF method, including reduced quadratic forms, infrastructural distance, and Gauss composition. We also present an analysis of our method, which has a computational complexity of $O \left( \exp \left( \frac{3}{\sqrt{8}} \sqrt{\ln N \ln \ln N} \right) \right)$, making it more efficient than the classical SQUFOF and CFRAC algorithms. Additionally, our algorithm is polynomial-time, provided knowledge of a (not too large) multiple of the regulator of $\mathbb{Q}(\sqrt{N})$.

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arXiv2024

Algebraic integers with continued fraction expansions containing palindromes and square roots with prescribed periods

We present a characterization of the algebraic integers with continued fraction expansions of the form $[a_0, \overline{a_1, \ldots, a_n, s}]$, where $(a_1, \ldots, a_n)$ is a palindrome and $s \in \mathbb{N}_{\geq 1}$. In particular, we focus on the special case where $(a_1, \ldots, a_n) = (m, \ldots, m)$, providing a detailed characterizations of the corresponding algebraic integers and $s$ in terms of Fibonacci polynomials. Then, we derive new expansions of square roots of integers with these periods, given $m$ and $n$. Moreover, we explicitly determine the fundamental solutions of both positive and negative Pell's equations corresponding to this family of integers.

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