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  "body": "\n## Definition\n\nLet $K$ be a field with a [[valuation|B85T5VKR3M]] $\\operatorname{val} \\colon K^\\times \\to \\R$, and let $K[x_1^{\\pm 1}, \\dots, x_n^{\\pm 1}]$ be the ring of Laurent polynomials.\nFor $f = \\sum_{u \\in \\Z^n} c_u x^u$ with finitely many nonzero $c_u$, the *tropicalization* of $f$ is the function $\\operatorname{trop}(f) \\colon \\R^n \\to \\R$,\n$$\\operatorname{trop}(f)(w) = \\min_{u : c_u \\neq 0} \\big( \\operatorname{val}(c_u) + u \\cdot w \\big).$$\nThe *tropical hypersurface* of $f \\neq 0$ is the set $\\operatorname{Trop}(f)$ of all $w \\in \\R^n$ at which this minimum is attained for at least two exponents $u$.\nFor an [[ideal|096Z9MP2ME]] $I$ of $K[x_1^{\\pm 1}, \\dots, x_n^{\\pm 1}]$, the *tropical variety* of $I$ is\n$$\\operatorname{Trop}(I) = \\bigcap_{f \\in I,\\ f \\neq 0} \\operatorname{Trop}(f).$$\n\n## Remarks\n\nFundamental theorem: let $K$ be [[algebraically closed|02SJG1HC32]] with $\\operatorname{val}$ nontrivial.\nThen $\\operatorname{Trop}(I)$ is the closure in $\\R^n$ of the points $(\\operatorname{val}(z_1), \\dots, \\operatorname{val}(z_n))$, for $z$ a zero of $I$ in the [[torus|66FQFAJNS4]] $(K^\\times)^n$.\nA typical such field is the field of [[Puiseux series|SENTYNSKSF]] over $\\C$, with the valuation given by the lowest exponent.\nIf moreover $I$ is prime and its zero set has dimension $d$, then $\\operatorname{Trop}(I)$ is the support of a pure $d$-dimensional polyhedral complex.\n\nExample: let $f = x + y + 1$ with all coefficients of valuation $0$.\nThen $\\operatorname{trop}(f)(w_1, w_2) = \\min(w_1, w_2, 0)$.\nThe minimum is attained twice exactly on the three rays $\\set{w_1 = w_2 \\le 0}$, $\\set{w_1 = 0 \\le w_2}$ and $\\set{w_2 = 0 \\le w_1}$, which form the tropical line.\nNon-example: the point $(1, 1)$ is not on it, since there $\\min(1, 1, 0) = 0$ is attained only by the constant term.\n\nThe intersection must run over all of $I$: the intersection of $\\operatorname{Trop}(f)$ over a set of generators of $I$ can be strictly larger than $\\operatorname{Trop}(I)$.\nA finite generating set for which equality holds is a tropical basis.\n",
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