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Mahler Measure

The Mahler measure of a polynomial P(x_1,x_2,...,x_k) is defined by

M_k(P)=exp[int_0^1...int_0^1ln|P(e^(2piit_1),...,e^(2piit_k))|dt_1...dt_k].
(1)

Using Jensen's formula, it can be shown that for P(x)=aproduct_(i=1)^(n)(x-alpha_i),

M_1(P)=|a|product_(i=1)^nmax{1,|alpha_i|}
(2)

(Borwein and Erdélyi 1995, p. 271).

Specific cases are given by

M_1(ax+b)=max{|a|,|b|}
(3)
M_2(1+x+y)=M_1(max{1,|1+x|})
(4)
M_2(1+x+y-xy)=M_1(max{|1-x|,|1+x|})
(5)

(Borwein and Erdélyi 1995, p. 272).

A product of cyclotomic polynomials has Mahler measure 1. The Mahler measure of an integer polynomial in k variables gives the topological entropy of a Z^k-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.

See also

Baernstein Quasi-Norm Monotonicity Conjecture, Jensen's Formula, Lehmer's Mahler Measure Problem, Polynomial Norm

This entry contributed by Kevin O'Bryant

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References

Borwein, P. and Erdélyi, T. "Mahler's Measure." §5.3.E.4 in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, pp. 271-272, 1995.Everest, G. and Ward, T. Heights of Polynomials and Entropy in Algebraic Dynamics. London, England: Springer-Verlag, 1999.

Referenced on Wolfram|Alpha

Mahler Measure

Cite this as:

Weisstein, Eric W., with contributions by Kevin O'Bryant. "Mahler Measure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MahlerMeasure.html

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