Tesi etd-06242025-144404 |
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Tipo di tesi
Tesi di laurea magistrale
Autore
MARZENTA, GIOVANNI
URN
etd-06242025-144404
Titolo
Effective results for families of norm form equations
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Prof. Zannier, Umberto
Parole chiave
- diophantine equations
- families of norm form equations
- norm form equations
- thue equations
Data inizio appello
18/07/2025
Consultabilità
Completa
Riassunto
This thesis focuses on families of norm form equations. We introduce the more traditional methods used to study this kind of Diophantine equations, namely Skolem's p-adic method and the linear forms in logarithms. In particular, we apply Skolem’s method to solve a parametric norm form equation in three variables of degree five.
We then turn to a recent approach developed by Amoroso, Masser, and Zannier, which allows to study families of norm form equations depending on an integer parameter t. We apply this method in order to effectively solve some families of norm form equations. In particular, we study certain generalizations of the equation previously solved via Skolem's method. We provide two alternative strategies for detecting functional solutions. Additionally, we present an alternative resolution of a family of Thue equations of degree four previously studied by Pethö.
The main hypotheses required to apply the method of Amoroso, Masser, and Zannier are closely related to the structure of the group of units of the coordinate ring of a curve associated to the given family of norm form equations. In the final part of the thesis, we study these unit groups, proving that they are finitely generated modulo constants and deriving formulas for their ranks. We also provide explicit results and effective algorithms for computing some minimal sets of generators for such groups.
We then turn to a recent approach developed by Amoroso, Masser, and Zannier, which allows to study families of norm form equations depending on an integer parameter t. We apply this method in order to effectively solve some families of norm form equations. In particular, we study certain generalizations of the equation previously solved via Skolem's method. We provide two alternative strategies for detecting functional solutions. Additionally, we present an alternative resolution of a family of Thue equations of degree four previously studied by Pethö.
The main hypotheses required to apply the method of Amoroso, Masser, and Zannier are closely related to the structure of the group of units of the coordinate ring of a curve associated to the given family of norm form equations. In the final part of the thesis, we study these unit groups, proving that they are finitely generated modulo constants and deriving formulas for their ranks. We also provide explicit results and effective algorithms for computing some minimal sets of generators for such groups.
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