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Newton polytope

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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 for the monomial . 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.

See also

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Sources

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  • 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 Mathematica. 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". Journal of the American Mathematical Society. 17 (1): 49–108. doi:10.1090/S0894-0347-03-00437-5. S2CID 14886953.
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