Polytopes requests

• packages: lrs, cddlib, porta, 4ti2, polymake, coin/or
• 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()

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()