Polytopes requests
packages: lrs, cddlib, porta, 4ti2, polymake, coin/or, http://www.math.u-bordeaux1.fr/~enge/Vinci.html
- optimal performance: important algorithms are reverse search (as in lrs, uses less memory), double description (track the duals, as in cdd and 4ti2)
- optimization: linear and integer programming (coin/or), semidefinite programming (any good software for this?)
- combinatorial aspects
- polymake puts a lot of these things together, but it does not build!
Bernstein's theorem
(this is coming from Daniel Erman).
R.<a,b>=QQ[] f1=a^2+a*b+b^2+1 f2=a*b^2+a^2*b+11 N1=f1.newton_polytope() N2=f2.newton_polytope() S=[N1,N2]
I would like to be able to compute the mixed volume of the collection of polytopes:
S.mixed_volume()
[[Note from Marshall Hampton: this is possible using the optional phc package:
from sage.interfaces.phc import phc phc.mixed_volume([f1,f2])
]]
The reason I want to do this is because I want to apply Bernstein's theorem to a polynomial system in affine space. So conceivably I'd like to ask:
F=[f1,f2] F.bernstein_bound()
In addition I'd like to be able to compute anything about N1 that can be done in polymake. For instance f-vector:
N1.f_vector()
How to deal with polyhedral fans ?
(from F. Chapoton)
I would like to work with polyhedral cones and fans (with toric geometry in mind), for instance
cone1=PolyhedralCone([[1,0,0],[0,1,0],[0,0,1]]) cone2=PolyhedralCone([[-1,0,0],[0,1,0],[0,0,1]]) fan=PolyhedralFan([cone1,cone2])
I would also need a conical convex hull algorithm
cone3=ConicalHull([[1,0,0],[0,1,0],[0,0,1],[1,1,1]])
should return the cone
PolyhedralCone([[1,0,0],[0,1,0],[0,0,1]])
Some support for fans is contained in GroebnerFan, but this would be better if separated clearly.
My long-term aim would be to compute the cone of strictly convex support functions of a complete fan.
fan.ample_cone()