Newton polytope information

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 $\mathbf {x} =(x_{1},\ldots ,x_{n})$ 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}\geq 0\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.

Newton polytope information

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