arXiv subjects

Marta Dujella

Publications and source records attributed to Marta Dujella.

2 recordsLinked to original sources

arXiv2023

Uniform Bounds for the Number of Rational Points of Bounded Height on Certain Elliptic Curves

Let $E$ be an elliptic curve defined over a number field $k$ and $\ell$ a prime integer. When $E$ has at least one $k$-rational point of exact order $\ell$, we derive a uniform upper bound $\exp(C \log B / \log \log B)$ for the number of points of $E(k)$ of (exponential) height at most $B$. Here the constant $C = C(k)$ depends on the number field $k$ and is effective. For $\ell = 2$ this generalizes a result of Naccarato which applies for $k=\mathbb{Q}$. We follow methods previously developed by Bombieri and Zannier and further by Naccarato, with the main novelty being the application of Rosen's result on bounding $\ell$-ranks of class groups in certain extensions, which is derived using relative genus theory.

math.NT↗

arXiv2023

Transcendence measure of $e^{1/n}$

For a given transcendental number $ξ$ and for any polynomial $P(X)=: λ_0+\cdots+λ_k X^k \in \mathbb{Z}[X]$, we know that $ P(ξ) \neq 0.$ Let $k \geq 1$ and $ω(k, H)$ be the infimum of the numbers $r > 0$ satisfying the estimate $$ \left|λ_0+λ_1 ξ+λ_2 ξ^{2}+ \ldots +λ_kξ^{k}\right| > \frac{1}{H^r}, $$ for all $(λ_0, \ldots ,λ_k)^T \in \mathbb{Z}^{k+1}\setminus\{\overline{0}\}$ with $\max_{1\le i\le k} \{|λ_i|\} \le H$. Any function greater than or equal to $ω(k, H)$ is a {\it transcendence measure of $ξ$}. In this article, we find out a transcendence measure of $ e^{1/n}$ which improves a result proved by Mahler(\cite{Mahler}) in 1975.

math.NT↗