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  "body": "\n## Summary\n\nThe Newton polytope of a polynomial in several variables is the [[convex hull|0FQSDC2EQ2]] of the exponent vectors of its monomials with nonzero coefficient.\nIt records which monomials can dominate, much as the [[Newton polygon|H7QNCE2QZJ]] of a polynomial over a valued field does in one variable.\nThe Newton polytope of a product is the Minkowski sum of the Newton polytopes of the factors.\nMixed volumes of Newton polytopes count the solutions of generic polynomial systems by the [[Bernstein–Kushnirenko theorem|VP32769PVR]].\nNewton polytopes link polynomials with [[toric varieties|YWB981MEF2]] and [[tropical geometry|7YTC3BHK60]].\n\n## Explanations\n\n### L1 human\n\nWrite a polynomial in $x$ and $y$ and mark, for each term $c\\,x^a y^b$ with $c \\ne 0$, the point $(a, b)$ in the plane.\nThe smallest convex region containing all the marked points is the Newton polytope.\nFor $1 + x + y + xy$ it is the unit square, and for $1 + x^2 + y^3$ it is a triangle.\nIts shape predicts much about the polynomial, for example how many common zeros two generic polynomials with these shapes have.\n",
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