StatePolytope -- computes state polytopes of ideals

Description

StatePolytope computes state polytopes of ideals using the programs gfan, M2, and polymake. Specifically, it computes State_m(I) or State(I) as defined in Sturmfels's book Groebner bases and convex polytopes, page 14. There is also a function for testing GIT stability of an ideal in a polynomial ring with respect to action of the maximal torus scaling the variables.

We assume that the user has gfan version 0.3 or higher on his or her system. The default assumption is that the command to run gfan is gfan, but this can be changed in the package configuration if necessary. The user also needs polymake version 2.3-1 or higher installed on his or her system, and it must be run by this command. Finally, the user is responsible for making sure the ideal of study is supported by gfan. In version 0.3 this means the ideal must be in a polynomial ring over QQ or ZZ/p with p < 32749, and no variable's name should be a substring of another's.

Author

Dave Swinarski <[email protected]>

Version

This documentation describes version 1.2 of StatePolytope.

Source code

The source code from which this documentation is derived is in the file StatePolytope.m2. The auxiliary files accompanying it are in the directory StatePolytope/.

Exports

Functions and commands
- initialIdeals -- calls gfan and returns the list of initial ideals
- isStable -- determines whether the mth Hilbert point of I is GIT stable
- polymakeStatePolytope -- computes state polytopes of ideals
Methods
- "initialIdeals(Ideal)" -- see initialIdeals -- calls gfan and returns the list of initial ideals
- "isStable(ZZ,Ideal)" -- see isStable -- determines whether the mth Hilbert point of I is GIT stable
- polymakeStatePolytope(Ideal) -- computes the state polytope of an ideal
- polymakeStatePolytope(ZZ,Ideal) -- computes the mth state polytope of an ideal

For the programmer

The object StatePolytope is a package.