For a polynomial , the Mahler measure of is defined by
(1)
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Using Jensen's formula, it can be shown that for ,
(2)
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(Borwein and Erdélyi 1995, p. 271).
Specific cases are given by
(3)
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(4)
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(5)
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(Borwein and Erdélyi 1995, p. 272).
A product of cyclotomic polynomials has Mahler measure 1. The Mahler measure of an integer polynomial in variables gives the topological entropy of a -dynamical system canonically associated to the polynomial.
Lehmer's Mahler measure problem conjectures that a particular univariate polynomial has the smallest possible Mahler measure other than 1.