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# Research
After obtaining a bachelor degree in Trento in 2016 I have completed the
[ALGANT](https://algant.eu) Master Program (Milano and Leiden), with
special focus on number theory and algebraic geometry. I then moved on to
obtain a joint PhD at the universities of Luxembourg and Leiden, under
the supervision of [Antonella Perucca](http://antonellaperucca.net) and
[Peter Bruin](https://www.math.leidenuniv.nl/~pbruin).
I have collected in this page the list of papers I puplished during my PhD,
as well as theses (Bachelor, Master, PhD). See [my talks page](../talks)
for a list of slides and notes of my talks.
## Theses
* Phd thesis: *Kummer theory for commutative algebraic groups*.
University of Luxembourg and University of Leiden, 2022.
Supervisors: Antonella Perucca, Peter J. Bruin.
[Thesis: [pdf, 1.2Mb](phd-thesis.pdf);
Stellingen: [pdf, 123Kb](stellingen.pdf)]
* Master thesis: *The Brauer-Manin obstruction to strong approximation*.
University of Milan and University of Leiden, 2018.
Supervisor: Martin J. Bright.
[[pdf, 800Kb](master-thesis.pdf)]
* Bachelor thesis: *Gruppi di Galois e gruppi fondamentali*.
University of Trento, 2016.
Supervisor: Edoardo Ballico.
[[pdf, 444Kb](bachelor-thesis.pdf)]
## Publications and preprints
* Tronto, Sebastiano.
*Division in modules and Kummer theory.*
[Journal de Thórie des Nombres de Bordeaux (2025) - open access](https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1326)
* Lombardo, Davide; Tronto, Sebastiano.
*Some uniform bounds for elliptic curves over Q.*
[Pacific Journal of Mathematics (2022)](https://msp.org/pjm/2022/320-1/p06.xhtml).
* Lombardo, Davide; Tronto, Sebastiano.
*Effective Kummer theory for elliptic curves.*
International Mathematics Research Notices (2021).
* Hörmann, Fritz; Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano.
*Explicit Kummer theory for quadratic fields.*
JP Journal of Algebra, Number Theory and Applications (2021).
* Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano.
*The degree of Kummer extensions of number fields.*
International Journal of Number Theory (2021).
* Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano.
*Kummer Theory for number fields via entanglement groups.*
Manuscripta Mathematica (2021).
* Tronto, Sebastiano.
*Radical entanglement for elliptic curves.*
[arXiv preprint](https://arxiv.org/abs/2009.08298).
* Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano.
*Kummer theory for number fields.*
Proceedings of the Roman Number Theory Association (2020).
* Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano.
*Addendum to: Reductions of Algebraic integers.*
Journal of Number Theory (2020).
* Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano.
*Explicit Kummer theory for the rational numbers.*
International Journal of Number Theory (2020).
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