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

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

where we use the shorthand notation (x1,…xK)(n1,…nK) = (xn1
1,…xnK
K). Then the Newton polytope associated to f is the convex hull of the {ak}k; that is

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

The Newton polytope satisfies the following homomorphism-type property:

${\displaystyle \operatorname {Newt} (fg)=\operatorname {Newt} (f)+\operatorname {Newt} (g)}$

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. 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.