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 ${\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})}$ of variables and a finite family ${\displaystyle (\mathbf {a} _{k})_{k}}$ of pairwise distinct vectors from ${\displaystyle \mathbb {N} ^{n}}$ each encoding the exponents within a monomial, consider the multivariate polynomial

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

where we use the shorthand notation ${\displaystyle (x_{1},\ldots ,x_{n})^{(y_{1},\ldots ,y_{n})}}$ for the monomial ${\displaystyle x_{1}^{y_{1}}x_{2}^{y_{2}}\cdots x_{n}^{y_{n}}}$. Then the Newton polytope associated to ${\displaystyle f}$ is the convex hull of the vectors ${\displaystyle \mathbf {a} _{k}}$; that is

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

In order to make this well-defined, we assume that all coefficients ${\displaystyle c_{k}}$ are non-zero. 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.

• Toric varieties
• Hilbert scheme

• 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.

• 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.