Newton polytope

In mathematics, the Newton polytope is an integral polytope associated with a multivariate polynomial. It can be used to analyze the polynomial's behavior when specific variables are considered negligible relative to the others. Specifically, given a vector $$\mathbf{x}=(x_1,\ldots,x_n)$$ of variables and a finite family $$(\mathbf{a}_k)_k$$ of pairwise distinct vectors from $$\mathbb{N}^n$$ each encoding the exponents within a monomial, consider the multivariate polynomial


 * $$f(\mathbf{x})=\sum_k c_k\mathbf{x}^{\mathbf{a}_k}$$

where we use the shorthand notation $$(x_1,\ldots,x_n)^{(y_1,\ldots,y_n)}$$ for the monomial $$x_1^{y_1}x_2^{y_2}\cdots x_n^{y_n}$$. Then the Newton polytope associated to $$f$$ is the convex hull of the vectors $$\mathbf{a}_k$$; that is


 * $$\operatorname{Newt}(f)=\left\{\sum_k \alpha_k\mathbf{a}_k :\sum_k \alpha_k =1\;\&\;\forall j\,\,\alpha_j\geq0\right\}\!.$$

In order to make this well-defined, we assume that all coefficients $$c_k$$ are non-zero. The Newton polytope satisfies the following homomorphism-type property:
 * $$\operatorname{Newt}(fg)=\operatorname{Newt}(f)+\operatorname{Newt}(g)$$

where the addition is in the sense of Minkowski.

Newton polytopes are the central object of study in tropical geometry and characterize the Gröbner bases for an ideal.