Decomposing a Random Newton Polygon

Consider a Newton polygon for a set of points in the plane with convex hull . The construction of the polygon is slightly different from the traditional one as follows (which is why some use the term Newton polytope).
The point is associated with the monomial , and a set of points in the plane is associated with the polynomial , that is, the sum of the monomials corresponding to the points in the set. For example, is associated with . Note that coefficients do not matter.
A polynomial that can be factored into a product of polynomials of lower degree is called reducible. The Newton polygon of a reducible polynomial corresponds to the Minkowski sum of the Newton polygons of the factors of . If a Newton polygon corresponds to a reducible polynomial, then is called decomposable.
In this Demonstration, a random set of points is chosen inside the square of side length from 1 to 8. The program checks if the corresponding Newton polygon is decomposable or indecomposable by checking all possible decompositions based on the boundary of the polygon; in the latter case, the corresponding polynomial is irreducible over any field, that is, absolutely (or geometrically) irreducible.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


[1] S. Gao and A. G. B. Lauder, "Decomposition of Polytopes and Polynomials," Discrete and Computational Geometry, 26(1), 2001 pp. 89–104. doi:10.1007/s00454-001-0024-0.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+