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, let

f(\mathbf {x} )=\sum _{k}c_{k}\mathbf {x} ^{\mathbf {a} _{k}}

where we use the shorthand notation $(x 1,\dots x K) (n 1,\dots n K) = (x n 1 1,\dots x n K K)$ . Then the Newton polytope associated to $f$ is the convex hull of the ${a k} k$ ; that is

\operatorname {Newt} (f)=\left\{\sum _{k}\alpha _{k}\mathbf {a} _{k}:\sum _{k}\alpha _{k}=1\;\&\;\forall j\,\,\alpha _{j}\geq 0\right\}.

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.

Sources

Sturmfels, Bernd (1996). "2. The State Polytope". Gröbner Bases and Convex Polytopes. University Lecture Series. Vol. 8. Providence, RI: AMS. ISBN 0-8218-0487-1.
Monical, Cara; Tokcan, Neriman; Yong, Alexander (10 March 2017). "Newton polytopes in algebraic combinatorics". arXiv:1703.02583v2.
Shiffman, Bernard; Zelditch, Steve (18 September 2003). "Random polynomials with prescribed Newton polytopes" (PDF). Journal of the AMS. 17 (1): 49–108. Retrieved 28 September 2019.

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