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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\operatorname{Log} \\colon (\\C^\\times)^n \\to \\R^n$ be the map\n$$\\operatorname{Log}(z_1, \\dots, z_n) = (\\log \\abs{z_1}, \\dots, \\log \\abs{z_n}).$$\nLet $f$ be a nonzero Laurent polynomial in $z_1, \\dots, z_n$ with [[complex|09FTAET5Z4]] coefficients, and let $V_f = \\set{z \\in (\\C^\\times)^n : f(z) = 0}$.\nThe *amoeba* of $f$ is the set $\\mathcal{A}_f = \\operatorname{Log}(V_f) \\subseteq \\R^n$.\n\n## Remarks\n\nThe amoeba is a closed subset of $\\R^n$, and each [[connected component|0JG9W3KA6B]] of its complement is convex.\nLet $f$ be nonconstant.\nThe number of these components is at least the number of vertices of the [[Newton polytope|CYTTX8KHC8]] of $f$, and at most its number of lattice points.\n\nExample: let $f = 1 + z + w$.\nThe point $(0, 0)$ lies in $\\mathcal{A}_f$: the point $z = e^{2\\pi i/3}$, $w = e^{-2\\pi i/3}$ has $\\abs{z} = \\abs{w} = 1$ and $1 + z + w = 0$.\n\nNon-example: the point $(1, 0)$ does not lie in $\\mathcal{A}_f$.\nIt would need $\\abs{z} = e$, $\\abs{w} = 1$ and $z = -1 - w$, but $\\abs{1 + w} \\le 2 \u003c e$.\n\nIn one variable, $f(z) = z - 2$ has $V_f = \\set{2}$ and $\\mathcal{A}_f = \\set{\\log 2}$.\n",
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