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 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 (2019). "Newton polytopes in algebraic combinatorics". Selecta Math. New Series. 25 (5): 66. arXiv:1703.02583. doi:10.1007/s00029-019-0513-8. S2CID 53639491.
Shiffman, Bernard; Zelditch, Steve (18 September 2003). "Random polynomials with prescribed Newton polytopes" (PDF). Journal of the American Mathematical Society. 17 (1): 49–108. doi:10.1090/S0894-0347-03-00437-5. S2CID 14886953. Retrieved 28 September 2019.

External Links[]

Linking Groebner Bases and Toric Varieties
Rossi, Michele; Terracini, Lea (2020). "Toric varieties and Gröbner bases: the complete Q-factorial case". Applicable Algebra in Engineering, Communication and Computing. 31 (5–6): 461–482. arXiv:2004.05092. doi:10.1007/s00200-020-00452-w.

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Newton polytope

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