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

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

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.

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