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{{endash}}Newton diagram developed for the analysis of bivariant polynomials.

Given a vector $\mathbf{x}=(x_1,\ldots,x_n)$ of variables and a finite family $(\mathbf{a}_k)_k$ of pairwise distinct vectors from $\mathbb{N}^n$ 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 $(x_1,\ldots,x_n)^{(y_1,\ldots,y_n)}$ 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

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

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:

$\operatorname{Newt}(fg)=\operatorname{Newt}(f)+\operatorname{Newt}(g)$

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

{{cite book|title=Gröbner Bases and Convex Polytopes|last=Sturmfels|first=Bernd|authorlink=Bernd Sturmfels|series=University Lecture Series|volume=8|publisher=AMS|location=Providence, RI|year=1996|isbn=0-8218-0487-1|chapter=2. The State Polytope|url-access=registration|url=https://archive.org/details/grobnerbasesconv0000stur}}
{{cite journal|title=Newton polytopes in algebraic combinatorics|last1=Monical|first1=Cara|last2=Tokcan|first2=Neriman|last3=Yong|first3=Alexander|arxiv=1703.02583 |journal=Selecta Mathematica |series=New Series |volume=25 |year=2019 |issue=5 |page=66|doi=10.1007/s00029-019-0513-8|doi-access=free|s2cid=53639491}}
{{cite journal|title=Random polynomials with prescribed Newton polytopes|last1=Shiffman|first1=Bernard|last2=Zelditch|first2=Steve|journal=Journal of the American Mathematical Society |volume=17|number=1|pages=49–108|date=18 September 2003|doi=10.1090/S0894-0347-03-00437-5|doi-access=free|s2cid=14886953}}

External links

[https://people.bath.ac.uk/ac886/students/jaquelineFreeke.pdf Linking Groebner Bases and Toric Varieties]
{{cite journal |last1=Rossi |first1=Michele |last2=Terracini |first2=Lea |title=Toric varieties and Gröbner bases: the complete Q-factorial case |journal=Applicable Algebra in Engineering, Communication and Computing |date=2020 |volume=31 |issue=5–6 |pages=461–482 |arxiv=2004.05092 |doi=10.1007/s00200-020-00452-w |doi-access=free}}

Category:Algebraic geometry

Category:Polynomial functions

Category:Minkowski spacetime

Category:Polytopes

Newton polytope

See also

Sources

External links