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

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

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

Newt ⁡ ( f ) = { ∑ k α k a k : ∑ k α k = 1 & ∀ j α j ≥ 0 } . {\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 c k {\displaystyle c_{k}} are non-zero. The Newton polytope satisfies the following homomorphism-type property:

Newt ⁡ ( f g ) = Newt ⁡ ( f ) + Newt ⁡ ( g ) {\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.

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