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| | In [[mathematics]], the '''Mahler measure''' <math>M(p)</math> of a [[polynomial]] <math>p(x) \in \mathbb{C}[x]</math> with [[complex number|complex]] coefficients is |
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| | :<math>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).</math> |
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| | :<math> ||p||_{\tau} =\left( { \frac{1}{2\pi} \int_{0}^{2\pi} |p(e^{i\theta})|^\tau \, d\theta } \right)^{1/\tau} \,</math> |
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| | is the [[Lp norm|<math>L_\tau</math> norm]] of <math>p</math> (although this is not a true norm for values of <math>\tau < 1</math>). |
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| | :<math>M(p) = |a| \prod_{i=1}^{n}\max\{1,|\alpha_i|\}=|a|\prod_{|\alpha_i| \ge 1} |\alpha_i|.</math> |
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| | The Mahler measure of an [[algebraic number]] <math>\alpha</math> is defined as the Mahler measure of the [[Minimal polynomial (field theory)|minimal polynomial]] of <math>\alpha</math> over <math>\mathbb{Q}</math>. |
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| | * The '''Mahler measure''' is multiplicative, i.e. <math>M(p\,q) = M(p) \cdot M(q).</math> |
| | * (Kronecker's Theorem) If <math>p</math> is an irreducible monic integer polynomial with <math>M(p) = 1</math>, then either <math>p(z) = z,</math> or <math>p</math> is a [[cyclotomic polynomial]]. |
| | * [[Lehmer's conjecture]] asserts that there is a constant <math>\mu>1</math> such that if <math>p</math> is an irreducible integer polynomial, then either <math>M(p)=1</math> or <math>M(p)>\mu</math>. |
| | * The Mahler measure of a monic integer polynomial is a [[Perron number]]. |
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| | ==Higher Dimensional Mahler Measure== |
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| | The Mahler measure <math>M(p)</math> of a multi-variable polynomial <math>p(x_1,\ldots,x_n) \in \mathbb{C}[x_1,\ldots,x_n]</math> |
| | is defined similarly by the formula |
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| | *{{Springer|id=m120070}} |
| | *{{cite book | author=Peter Borwein | authorlink=Peter Borwein | title=Computational Excursions in Analysis and Number Theory | series=CMS Books in Mathematics | publisher=[[Springer-Verlag]] | year=2002 | isbn=0-387-95444-9 | pages=3, 15 }} |
| | *{{cite journal | author=J.L. Jensen | title=Sur un nouvel et important théorème de la théorie des fonctions | journal=[[Acta Mathematica]] | volume=22 | pages=359–364 | year=1899 | doi=10.1007/BF02417878 }} |
| | *{{cite book |author=[[Donald E. Knuth|Knuth, Donald E]] |chapter=4.6.2 Factorization of Polynomials |title=Seminumerical Algorithms |series=The Art of Computer Programming |volume=2 |edition=Third |location=Reading, Massachusetts |publisher=Addison-Wesley |year=1997 |pages=439–461, 678–691<!-- xiv+762 --> |
| | |isbn=0-201-89684-2}} |
| | *{{cite journal | author=M.J. Mossinghoff | title=Polynomials with Small Mahler Measure | journal=Mathematics of Computation | volume=67 | pages=1697–1706 | year=1998 | doi=10.1090/S0025-5718-98-01006-0 | issue=224 }} |
| | * {{cite book | first=Chris | last=Smyth | chapter=The Mahler measure of algebraic numbers: a survey | pages=322–349 | editor1-first=James | editor1-last=McKee | editor2-last=Smyth | editor2-first=Chris | title=Number Theory and Polynomials | series=London Mathematical Society Lecture Note Series | volume=352 | publisher=[[Cambridge University Press]] | year=2008 | isbn=978-0-521-71467-9 }} |
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