In November 2012, I started the Perfezionamento (a PhD) at Scuola Normale Superiore Pisa in Mathematics. I was very fortunate to have Prof. Umberto Zannier as my supervisor, and I submitted my thesis in Mai 2016.
I defended my thesis successfully in September 2016.
The title of my thesis is Specialization and Reduction of hyperelliptic continued fractions, it investigates the appearance of prime numbers in the denominators in continued fractions of square roots of polynomials.
Such phenomena are closely tied to reduction questions, which are investigated in detail here. The first parts deal with general aspects of hyperelliptic continued fractions, and could serve as a detailed introduction to the topic.
From 2013 to 2015 I tutored, together with Laura Capuano and Fabrizio Barroero the second year course of Prof. Zannier at SNS. This lecture treats miscellaneous topics in algebra, geometry and number theory and was held two years in sequence.
During my PhD, I spent quite some time doing calculations, or at least writing code to do calculations. In some cases, this has actually helped my research, but certainly it was a lot of fun.
I even attempted to write my own computer algebra system. Despite its limitations, it has still the nicest implementation for continued fractions of general Laurent series I wrote so far. And I have written a few of them, some in Mathematica.
For my research, I also built a library for computing and analysing hyperelliptic continued fractions (essentially continued fractions of the square root of a polynomial).The download includes all the necessary Sage/Python code and some example instruction on how to use the library. Please note an installation and basic knowledge of SageMath is required.
The main features of the library include: