newtonPolytope -- computes the Newton polytope of a polynomial

Synopsis

• Usage:

  P = newtonPolytope f

• Inputs:
  • f, a ring element
• Outputs:
  • P, a convex polyhedron

Description

The newtonPolytope of f is the convex hull of its exponent vectors in n-space, where n is the number of variables in the ring.

Consider the Vandermond determinant in 3 variables:

i1 : R = QQ[a,b,c]

o1 = R

o1 : PolynomialRing

i2 : f = (a-b)*(a-c)*(b-c)

      2       2    2     2       2      2
o2 = a b - a*b  - a c + b c + a*c  - b*c

o2 : R

If we compute the Newton polytope we get a hexagon in QQ^3.

i3 : P = newtonPolytope f

o3 = P

o3 : Polyhedron