Newton polytope

 ( Algebraic Geometry ) 

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 KruskalNewton diagram developed for the analysis of bivariant polynomials.

Given a vector $\backslash mathbf\{x\}=(x\_1,\backslash ldots,x\_n)$ of variables and a finite family $(\backslash mathbf\{a\}\_k)\_k$ of pairwise distinct vectors from $\backslash mathbb\{N\}^n$ each encoding the exponents within a monomial, consider the multivariate polynomial

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

where we use the shorthand notation $(x\_1,\backslash ldots,x\_n)^\{(y\_1,\backslash ldots,y\_n)\}$ for the monomial $x\_1^\{y\_1\}x\_2^\{y\_2\}\backslash cdots\; x\_n^\{y\_n\}$. Then the Newton polytope associated to $f$ is the convex hull of the vectors $\backslash mathbf\{a\}\_k$; that is

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: $$\backslash operatorname\{Newt\}(fg)=\backslash operatorname\{Newt\}(f)+\backslash operatorname\{Newt\}(g)$$ where the addition is in Minkowski sum.

Newton polytopes are the central object of study in tropical geometry and characterize the Gröbner bases for an ideal.

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See also

• Toric varieties
• Hilbert scheme

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Sources

• Bernd Sturmfels (1996). 9780821804872, AMS. . ISBN 9780821804872

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External links

• Linking Groebner Bases and Toric Varieties

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