NP -- the Newton polyhedron of a monomial ideal

Synopsis

Usage:

NP(I)
Inputs:
- I, an ideal,
Outputs:
- a convex polyhedron, 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

Ways to use NP :

NP(Ideal)

For the programmer

The object NP is a method function.

NP -- the Newton polyhedron of a monomial ideal

Synopsis

Description

See also

Ways to use NP :

For the programmer