Differences between revisions 7 and 9 (spanning 2 versions)
⇤ ← Revision 7 as of 2009-09-14 10:31:33 →
Size: 2205
Editor: FrédéricChapoton
Comment: link to polymake
← Revision 9 as of 2014-07-18 20:23:09 → ⇥
Size: 2162
Editor: chapoton
Comment:
Deletions are marked like this. Additions are marked like this.
Line 28: Line 28:
[[Note from Marshall Hampton: this is possible using the optional phc package:  [[Note from Marshall Hampton: this is possible using the optional phc package:
Line 52: Line 52:
(from F. Chapoton) I would like to work with polyhedral cones and fans (with toric geometry in mind).
Line 54: Line 54:
I would like to work with polyhedral cones and fans (with toric geometry in mind), for instance The following now works (sage 5.2):
Line 57: Line 57:
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])
Line 62: Line 62:
I would also need a conical convex hull algorithm The '''Cone''' function can take many generators
Line 64: Line 64:
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]])
Line 66: Line 66:
should return the cone and then compute the extremal rays
Line 68: Line 68:
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
}}

It is now possible (sage 6.2) compute the cone of strictly convex support functions of a complete fan.
{{{
ToricVariety(fan).Kaehler_cone()
Line 70: Line 79:

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()
}}}

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()

Another note from M. Hampton: I have a patch for computing face lattices and f-vectors that I am hoping to put up on trac this week.

How to deal with polyhedral fans ?

I would like to work with polyhedral cones and fans (with toric geometry in mind).

The following now works (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
}}

It is now possible (sage 6.2) compute the cone of strictly convex support functions of a complete fan.
{{{
ToricVariety(fan).Kaehler_cone()