Newton polytope

Summary

The Newton polytope of a polynomial in several variables is the convex hull of the exponent vectors of its monomials with nonzero coefficient. It records which monomials can dominate, much as the Newton polygon of a polynomial over a valued field does in one variable. The Newton polytope of a product is the Minkowski sum of the Newton polytopes of the factors. Mixed volumes of Newton polytopes count the solutions of generic polynomial systems by the Bernstein–Kushnirenko theorem. Newton polytopes link polynomials with toric varieties and tropical geometry.