The arithmetic and geometry of Salem numbers

By Eknath Ghate, E. Hironaka. 2001

60 citations recorded by the source corpus.

A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and arithmetic, and discuss the possible implications for the ‘minimization problem’. This is an old question in number theory which asks whether the set of Salem numbers is bounded away from 1.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d7cdfe1c67b2516ab599357332c93c801d52542f

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In Computational Theory and Mathematics

1. Christopher J. Leininger
2. Curtis T. McMullen
3. Michael W. Davis
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In Discrete Mathematics and Combinatorics

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In Mathematical Physics

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Citations in the corpus

• Seventy years of Salem numbers (2014, unknown): “The survey of Ghate and Hironaka [46] contains many applications of Salem numbers, for the period up to 1999.”
• On groups generated by two positive multi-twists: Teichmuller curves and Lehmer's number (2003, background): “In particular, an affirmative answer to the following (see [10] and [24]) would be a consequence of such an answer to Lehmer’s question.”
• Coxeter groups, Salem numbers and the Hilbert metric (2002, background): “See [ GH ] for a recent survey on this topic.”