Mahler's Measure, Lehmer's Conjecture, and Some Connections - I

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Mathematics (CMC)

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Mahler's measure of polynomials in one variable with complex coefficients is a function measuring the extent to which the roots of a polynomial are distributed outside of the unit circle. I will start by introducing Mahler's measure and stating a famous conjecture of D. H. Lehmer about the values it assumes when restricted to monic polynomials with integer coefficients. This conjecture has great significance in mathematics, especially in Number Theory and Ergodic Theory.

Lehmer's conjecture is closely connected to the so called minimization problem for Salem numbers. In fact, any insight into the size of the smallest possible Salem number would constitute a major step towards Lehmer's conjecture. Salem numbers appear as largest poles of the rational growth functions for certain Coxeter groups, and hence are related to the asymptotic growth rates of these groups. This fascinating connection, coming from the work of Cannon, E. Hironaka, and others, provides additional evidence in support of Lehmer's conjecture. I will review some results in this direction.

This will be an informal expository talk with all the necessary notation and background information provided.


This lecture was given during the Analysis Seminar at the Claremont Colleges in February 2008.

The second part of this lecture, by the same author, is "Mahler's measure, Lehmer's conjecture, and some connections - II" and was given during the Analysis Seminar at the Claremont Colleges in April 2008.

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© 2008 Lenny Fukshansky

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