In mathematics, the Newton polytope is an integral polytope associated with a multivariate polynomial that can be used in the asymptotic analysis of those polynomials. It is a generalization of the KruskalNewton diagram developed for the analysis of bivariant polynomials.
Given a vector of variables and a finite family of pairwise distinct vectors from each encoding the exponents within a monomial, consider the multivariate polynomial
where we use the shorthand notation for the monomial . Then the Newton polytope associated to is the convex hull of the vectors ; that is
In order to make this well-defined, we assume that all coefficients are non-zero. The Newton polytope satisfies the following homomorphism-type property:
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.