# NP -- the Newton polyhedron of a monomial ideal

## Synopsis

• Usage:
NP(I)
• Inputs:
• Outputs:
• , the Newton polyhedron of the monomial ideal I

## Description

Given a monomial ideal I in $k[x_1,\dots,x_d]$, the convex hull in $\mathbb{R}^d$ of the set of exponents of all monomials in I is called the Newton polyhedron of I.

 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : I = ideal"x2,y3,yz" 2 3 o2 = ideal (x , y , y*z) o2 : Ideal of R i3 : P = NP I o3 = {ambient dimension => 3 } dimension of lineality space => 0 dimension of polyhedron => 3 number of facets => 5 number of rays => 3 number of vertices => 3 o3 : Polyhedron

Note that a monomial is in the integral closure of I if and only if its exponent vector is in NP(I).

 i4 : J = integralClosure(I,1) 2 3 2 o4 = ideal (y*z, x , y , x*y ) o4 : Ideal of R i5 : P == NP J o5 = true