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  "body": "\n## Definition\n\nLet $K$ be a [[field|04C5ZBG1WT]] and let $f = \\sum_{a \\in \\Z^n} c_a x^a$ be a Laurent polynomial in $x_1, \\dots, x_n$ over $K$, where $x^a = x_1^{a_1} \\cdots x_n^{a_n}$ and only finitely many $c_a$ are nonzero.\nThe *support* of $f$ is $\\operatorname{supp}(f) = \\set{a \\in \\Z^n : c_a \\ne 0}$.\nThe *Newton polytope* of $f$ is the [[convex hull|0FQSDC2EQ2]]\n$$\\operatorname{Newt}(f) = \\operatorname{conv}(\\operatorname{supp}(f)) \\subseteq \\R^n.$$\n\n## Remarks\n\nThe Newton polytope is a lattice polytope: its vertices lie in $\\Z^n$, and each vertex is the exponent of a monomial of $f$.\nFor nonzero $f, g$, one has $\\operatorname{Newt}(fg) = \\operatorname{Newt}(f) + \\operatorname{Newt}(g)$, the Minkowski sum.\n\nExample: for $f = 1 + x + y + xy$ the support is $\\set{(0,0), (1,0), (0,1), (1,1)}$ and $\\operatorname{Newt}(f)$ is the unit square.\nIndeed $f = (1 + x)(1 + y)$, and the unit square is the Minkowski sum of the segments $[0,1] \\times \\set{0}$ and $\\set{0} \\times [0, 1]$.\n\nNon-example: for $f = x^2 - (x - 1)(x + 1) = 1$, the support is $\\set{0}$, not $\\set{0, 1, 2}$.\nThe Newton polytope uses the monomials that remain after simplification, so $\\operatorname{Newt}(f)$ is the single point $0$ and not the segment $[0, 2]$.\n\nIn one variable, the Newton polytope of $f$ is the segment from the lowest to the highest exponent.\n",
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