Newton polytope

rdf:langString Newton polytope

rdf:langString 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 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 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.