Newton polytope

Remarks

The Newton polytope is a lattice polytope: its vertices lie in $\Z^n$, and each vertex is the exponent of a monomial of $f$. For nonzero $f, g$, one has $\operatorname{Newt}(fg) = \operatorname{Newt}(f) + \operatorname{Newt}(g)$, the Minkowski sum.

Example: 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. Indeed $f = (1 + x)(1 + y)$, and the unit square is the Minkowski sum of the segments $[0,1] \times \set{0}$ and $\set{0} \times [0, 1]$.

Non-example: for $f = x^2 - (x - 1)(x + 1) = 1$, the support is $\set{0}$, not $\set{0, 1, 2}$. The Newton polytope uses the monomials that remain after simplification, so $\operatorname{Newt}(f)$ is the single point $0$ and not the segment $[0, 2]$.

In one variable, the Newton polytope of $f$ is the segment from the lowest to the highest exponent.