The Newton polytope of a polynomial is the convex hull
of its exponent vectors. It records which monomials are present, discarding the coefficients’ values. Sums correspond to unions of vertex sets, and products to Minkowski sums, so that .
Lifting each exponent to height and taking the lower faces of projects to a regular subdivision of , also called coherent. The coefficients thereby choose how the polytope is cut into cells. Generic heights give a triangulation, whereas special heights merge cells.
This subdivision is dual to the tropical hypersurface of . Cells of the subdivision correspond to complementary regions where a single monomial dominates, edges to the facets of the tropical variety, and interior edges to its ridges. So the Newton polytope and its lift form the combinatorial blueprint of the piecewise-linear shape, the lever deciding which facets appear and how they meet.