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| = Polytopes requests = See [[https://wiki.sagemath.org/OptiPolyGeom]] for a more recent page. * packages: lrs, cddlib, porta, 4ti2, [[https://www.polymake.org/doku.php|polymake]], coin/or, [[http://www.math.u-bordeaux1.fr/~enge/Vinci.html|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. {{{ sage: fan = Fan([cone3]) sage: ToricVariety(fan).Kaehler_cone() }}} |
