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  "body": "\n## Definition\n\nLet $f \\in \\C[x]$ be a nonzero polynomial.\nBy the [[fundamental theorem of algebra|0JP0W07Y9A]], write $f = a (x - \\alpha_1) \\cdots (x - \\alpha_n)$ with $a \\in \\C \\setminus \\set{0}$ and [[roots|00EHX6FWC0]] $\\alpha_1, \\dots, \\alpha_n \\in \\C$.\nThe Mahler measure of $f$ is\n$$M(f) = \\abs{a} \\prod_{i=1}^{n} \\max(1, \\abs{\\alpha_i}).$$\n\n## Remarks\n\nBy Jensen's formula, $M(f) = \\exp\\left( \\int_0^1 \\log \\abs{f(e^{2 \\pi i \\theta})} \\, d\\theta \\right)$.\nIt is multiplicative: $M(f g) = M(f) M(g)$.\nExample: $f = 2x - 1 = 2(x - \\frac{1}{2})$ gives $M(f) = 2 \\cdot \\max(1, \\frac{1}{2}) = 2$.\nNon-example of a common confusion: $M(f)$ is not the largest absolute value of a coefficient; for $f = (x + 1)^2 = x^2 + 2x + 1$ that number is $2$, while $M(f) = 1$.\nIf $f \\in \\Z[x]$ is nonzero, then $M(f) \\ge 1$.\nBy a theorem of Kronecker, a nonzero $f \\in \\Z[x]$ has $M(f) = 1$ if and only if $f$ is $\\pm$ a product of a power of $x$ and [[cyclotomic polynomials|08KEYFAH66]].\n\n## Explanations\n\n### L1 human\n\nThe Mahler measure looks at where the roots are.\nRoots inside the unit circle are ignored, and roots outside count with their size.\n",
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