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 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: