Mahler measure

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In mathematics, the Mahler measure M(p) of a polynomial p(x)\in {\mathbb {C}}[x] with complex coefficients is

M(p)=\lim _{{\tau \rightarrow 0}}\|p\|_{{\tau }}=\exp \left({\frac {1}{2\pi }}\int _{{0}}^{{2\pi }}\ln(|p(e^{{i\theta }})|)\,d\theta \right).

Here

||p||_{{\tau }}=\left({{\frac {1}{2\pi }}\int _{{0}}^{{2\pi }}|p(e^{{i\theta }})|^{\tau }\,d\theta }\right)^{{1/\tau }}\,

is the L_{\tau } norm of p (although this is not a true norm for values of \tau <1).

It can be shown that if

p(z)=a(z-\alpha _{1})(z-\alpha _{2})\cdots (z-\alpha _{n})

then

M(p)=|a|\prod _{{i=1}}^{{n}}\max\{1,|\alpha _{i}|\}=|a|\prod _{{|\alpha _{i}|\geq 1}}|\alpha _{i}|.

The Mahler measure of an algebraic number \alpha is defined as the Mahler measure of the minimal polynomial of \alpha over {\mathbb {Q}}.

The measure is named after Kurt Mahler.

Properties

  • The Mahler measure is multiplicative, i.e. M(p\,q)=M(p)\cdot M(q).
  • (Kronecker's Theorem) If p is an irreducible monic integer polynomial with M(p)=1, then either p(z)=z, or p is a cyclotomic polynomial.
  • Lehmer's conjecture asserts that there is a constant \mu >1 such that if p is an irreducible integer polynomial, then either M(p)=1 or M(p)>\mu .
  • The Mahler measure of a monic integer polynomial is a Perron number.

Higher Dimensional Mahler Measure

The Mahler measure M(p) of a multi-variable polynomial p(x_{1},\ldots ,x_{n})\in {\mathbb {C}}[x_{1},\ldots ,x_{n}] is defined similarly by the formula

M(p)=\exp \left({\frac {1}{(2\pi )^{n}}}\int _{{0}}^{{2\pi }}\int _{{0}}^{{2\pi }}\cdots \int _{{0}}^{{2\pi }}\ln {\Bigl (}{\bigl |}p(e^{{i\theta _{1}}},e^{{i\theta _{2}}},\ldots ,e^{{i\theta _{n}}}){\bigr |}{\Bigr )}\,d\theta _{1}\,d\theta _{2}\cdots d\theta _{n}\right).

Multi-variable Mahler measures have been shown, in some cases, to be related to special values of zeta-functions and L-functions.

See also

References

  • Hazewinkel, Michiel, ed. (2001), "m120070", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
  • Peter Borwein (2002). Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics. Springer-Verlag. pp. 3, 15. ISBN 0-387-95444-9. 
  • J.L. Jensen (1899). "Sur un nouvel et important théorème de la théorie des fonctions". Acta Mathematica 22: 359–364. doi:10.1007/BF02417878. 
  • Knuth, Donald E (1997). "4.6.2 Factorization of Polynomials". Seminumerical Algorithms. The Art of Computer Programming 2 (Third ed.). Reading, Massachusetts: Addison-Wesley. pp. 439–461, 678–691. ISBN 0-201-89684-2. 
  • M.J. Mossinghoff (1998). "Polynomials with Small Mahler Measure". Mathematics of Computation 67 (224): 1697–1706. doi:10.1090/S0025-5718-98-01006-0. 
  • Smyth, Chris (2008). "The Mahler measure of algebraic numbers: a survey". In McKee, James; Smyth, Chris. Number Theory and Polynomials. London Mathematical Society Lecture Note Series 352. Cambridge University Press. pp. 322–349. ISBN 978-0-521-71467-9. 

External links

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