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 [math]\displaystyle{ \mathbf{x}=(x_1,\ldots,x_n) }[/math] of variables and a finite family [math]\displaystyle{ (\mathbf{a}_k)_k }[/math] of pairwise distinct vectors from [math]\displaystyle{ \mathbb{N}^n }[/math] each encoding the exponents within a monomial, consider the multivariate polynomial

[math]\displaystyle{ f(\mathbf{x})=\sum_k c_k\mathbf{x}^{\mathbf{a}_k} }[/math]

where we use the shorthand notation [math]\displaystyle{ (x_1,\ldots,x_n)^{(y_1,\ldots,y_n)} }[/math] for the monomial [math]\displaystyle{ x_1^{y_1}x_2^{y_2}\cdots x_n^{y_n} }[/math]. Then the Newton polytope associated to [math]\displaystyle{ f }[/math] is the convex hull of the vectors [math]\displaystyle{ \mathbf{a}_k }[/math]; that is

[math]\displaystyle{ \operatorname{Newt}(f)=\left\{\sum_k \alpha_k\mathbf{a}_k :\sum_k \alpha_k =1\;\&\;\forall j\,\,\alpha_j\geq0\right\}\!. }[/math]

In order to make this well-defined, we assume that all coefficients [math]\displaystyle{ c_k }[/math] are non-zero. The Newton polytope satisfies the following homomorphism-type property:

[math]\displaystyle{ \operatorname{Newt}(fg)=\operatorname{Newt}(f)+\operatorname{Newt}(g) }[/math]

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

• Toric varieties
• Hilbert scheme

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. https://archive.org/details/grobnerbasesconv0000stur. 
• Monical, Cara; Tokcan, Neriman; Yong, Alexander (2019). "Newton polytopes in algebraic combinatorics". Selecta Mathematica. New Series 25 (5): 66. doi:10.1007/s00029-019-0513-8. 
• 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. 

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. doi:10.1007/s00200-020-00452-w.