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The Schinzel-Zassenhaus conjecture SemesterWiSe 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
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
BSem, MV, MSem, LA-GySem ECTS 4,5 |