Newton polytope

In mathematics, the Newton polytope is an integral polytope associated with a multivariate polynomial that can be used in the asymptotic analysis of those polynomials. It is a generalization of the Kruskal–Newton diagram developed for the analysis of bivariant polynomials.

Given a vector $\mathbf{𝐱}=(x_1,\dots ,x_n)$ of variables and a finite family $({\mathbf{𝐚}}_k{)}_k$ of pairwise distinct vectors from ${\mathbb{\mathbb{N}}}^n$ each encoding the exponents within a monomial, consider the multivariate polynomial

$$f(\mathbf{𝐱})=\sum _k{c}_k{\mathbf{𝐱}}^{{\mathbf{𝐚}}_k}$$

where we use the shorthand notation $(x_1,\dots ,x_n{)}^{(y_1,\dots ,y_n)}$ for the monomial $x_1^{y_1}{x}_2^{y_2}\cdots x_n^{y_n}$. Then the Newton polytope associated to $f$ is the convex hull of the vectors ${\mathbf{𝐚}}_k$; that is

$$\mathrm{Newt}(f)=\{\sum _k{\alpha }_k{\mathbf{𝐚}}_k:\sum _k{\alpha }_k=1\mathrm{\&}\mathrm{\forall }j{\alpha }_j\ge 0\}.$$

In order to make this well-defined, we assume that all coefficients $c_k$ are non-zero. The Newton polytope satisfies the following homomorphism-type property: $$\mathrm{Newt}(fg)=\mathrm{Newt}(f)+\mathrm{Newt}(g)$$ 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.

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.