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Fakultät für Mathematik Universität Regensburg
The Schinzel-Zassenhaus conjecture
Semester
WiSe 2023 / 24

Lecturer
Walter Gubler und Debam Biswas

Type of course (Veranstaltungsart)
Seminar

German title
Die Schinzel-Zassenhaus Vermutung

Contents
The former Schinzel-Zassenhaus conjecture, proposed in the sixties of last century, states that the complex roots of a polynomial with integer coefficients are either all located on the unit circle or at least one of them has a sufficiently large absolute value in terms of the degree of the polynomial. The statement can also be viewed as a description of the size of the Galois conjugates of an algebraic integer, and hence has implications on its height and on the Mahler measure of its minimal polynomial.
In this seminar, we aim at understanding the recent (and surprisingly simple!) proof of the conjecture by Dimitrov. In the process, we introduce and study notions which are central in diophantine geometry, like the one of height and of Mahler measure, and also investigate some more analytical constructions, such as Pólya's theorem on the rationality of series and some potential theory in the complex plane.

Literature
  • Enrico Bombieri, Walter Gubler, Heights in Diophantine Geometry, volume 4 in the series ''Cambridge New Mathematical Monographs'' (2006);
  • Vesselin Dimitrov, A proof of the Schinzel-Zassenhaus conjecture on polynomials, preprint on arXiv:1912.12545 (2019).


Recommended previous knowledge
Some basic complex analysis and Galois theory. A previous exposure to algebraic number theory will be helpful, but not necessary.

Time/Date
Mon 10-12

Location
M103

Registration
  • Organisational meeting/distribution of topics: Thursday 20.07, 14-15, room M104
  • Registration for course work/examination/ECTS: FlexNow
Course work (Studienleistungen)
  • Presentation: Giving a seminar talk of roughly 90 minutes
Examination (Prüfungsleistungen)
  • Detailed written report of the seminar talk
Modules
BSem, MV, MSem, LA-GySem

ECTS
4,5