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Comment: link to polymake
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Comment: + some things that exist
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| [[Note from Marshall Hampton: this is possible using the optional phc package: | [[Note from Marshall Hampton: this is possible using the optional phc package: |
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| I would like to work with polyhedral cones and fans (with toric geometry in mind), for instance | I would like to work with polyhedral cones and fans (with toric geometry in mind). The following now works (2012-09, sage 5.2) |
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| 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]) |
sage: cone1 = Cone([[1,0,0],[0,1,0],[0,0,1]]) sage: cone2 = Cone([[-1,0,0],[0,1,0],[0,0,1]]) sage: fan = Fan([cone1,cone2]) |
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| I would also need a conical convex hull algorithm | The '''Cone''' function can take many generators |
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| cone3=ConicalHull([[1,0,0],[0,1,0],[0,0,1],[1,1,1]]) | sage: cone3 = Cone([[1,0,0],[0,1,0],[0,0,1],[1,1,1]]) |
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| should return the cone | and then compute the extremal rays |
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| PolyhedralCone([[1,0,0],[0,1,0],[0,0,1]]) | sage: cone3.rays() N(0, 0, 1), N(0, 1, 0), N(1, 0, 0) in 3-d lattice N |
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Some support for fans is contained in GroebnerFan, but this would be better if separated clearly. |
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There already exists the attribute '''is_complete''' for fans. |
Polytopes requests
packages: lrs, cddlib, porta, 4ti2, polymake, coin/or, Vinci
- 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).
The following now works (2012-09, sage 5.2)
sage: cone1 = Cone([[1,0,0],[0,1,0],[0,0,1]]) sage: cone2 = Cone([[-1,0,0],[0,1,0],[0,0,1]]) sage: fan = Fan([cone1,cone2])
The Cone function can take many generators
sage: cone3 = Cone([[1,0,0],[0,1,0],[0,0,1],[1,1,1]])
and then compute the extremal rays
sage: cone3.rays() N(0, 0, 1), N(0, 1, 0), N(1, 0, 0) in 3-d lattice N
My long-term aim would be to compute the cone of strictly convex support functions of a complete fan.
fan.ample_cone()
There already exists the attribute is_complete for fans.