Orthogonal decomposition of the space of algebraic numbers and Lehmer's problem

Paul Fili; Zachary Miner

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arXiv · 0911.1975

Orthogonal decomposition of the space of algebraic numbers and Lehmer's problem

Paul Fili·Zachary Miner

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Abstract

We introduce vector space norms associated to the Mahler measure by using the L^p norm versions of the Weil height recently introduced by Allcock and Vaaler. In order to do this, we determine orthogonal decompositions of the space of algebraic numbers modulo torsion by Galois field and degree. We formulate L^p Lehmer conjectures involving lower bounds on these norms and prove that these new conjectures are equivalent to their classical counterparts, specifically, the classical Lehmer conjecture in the p = 1 case and the Schinzel-Zassenhaus conjecture in the p = infinity case.

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Paul Fili, Zachary Miner. 2009-11-10. Orthogonal decomposition of the space of algebraic numbers and Lehmer's problem. https://arxiv.org/abs/0911.1975

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