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Number Theory Seminar

A solution of the Schinzel-Zassenhaus conjecture and a height gap theorem on holonomic sequences

Vesselin Dimitrov, University of Toronto

Location:  URL:
Date & time: Tuesday, 21 April 2020 at 2:00PM - 3:00PM

Abstract: I will explain the classical arithmetic rationality criterion of Polya and Bertrandias, as well as a theorem of Dubinin in potential theory, and combine them to derive a solution of the Schinzel-Zassenhaus conjecture: a monic integer irreducible polynomial of degree \(n > 1\) is either cyclotomic, or else has at least one complex root outside of the disk of radius \(2^{1/(4n)} > 1 + \frac{\log{2}}{4n}\). The Chowla-Blanksby hypothetical extension to non-monic polynomials (which was originally motivated by Turan's power sums method), as well as the full Lehmer problem on Mahler measure (specifically its Salem case), seemingly remain outside of the scope of our proposed method. Instead, we discover a different generalization where the Schinzel-Zassenhaus problem fits naturally as a gap theorem on the exponential growth rates of integer holonomic sequences.

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