polymakeStatePolytope(Ideal) -- computes the state polytope of an ideal

Synopsis

Function: polymakeStatePolytope
Usage:

statePolytope(I)
Inputs:
- an ideal, the ideal
Outputs:
- a list, the list of vertices of the state polytope

Description

See Sturmfels's book Groebner bases and convex polytopes, page 14 for the definition of State(I). (The difference between this and State_m(I) is that for all sufficiently large m, State_m(I) does not distinguish between initial ideals which have the same saturation with regard to the irrelevant ideal, whereas in State(I), these are separated.)

i1 : R = QQ[a..d];

i2 : I = ideal(a*c-b^2,a*d-b*c,b*d-c^2);  

o2 : Ideal of R

i3 : polymakeStatePolytope(I)
polymake: used package cdd
  cddlib
  Implementation of the double description method of Motzkin et al.
  Copyright by Komei Fukuda.
  http://www-oldurls.inf.ethz.ch/personal/fukudak/cdd_home/

VERTICES
1 11 7 7 11
1 7 15 3 11
1 4 18 6 8
1 3 18 9 6
1 11 3 15 7
1 8 6 18 4
1 6 9 18 3
1 3 15 15 3


o3 = {{11, 7, 7, 11}, {7, 15, 3, 11}, {4, 18, 6, 8}, {3, 18, 9, 6}, {11, 3,
     ------------------------------------------------------------------------
     15, 7}, {8, 6, 18, 4}, {6, 9, 18, 3}, {3, 15, 15, 3}}

o3 : List

Ways to use this method: