Finding a lifting for a given subdivision

[EnisKaya]

The following code, which is from Tropical Computations in polymake by Hampe-Joswig,

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$points = [[0,0,4],[1,0,3],[0,1,3],[2,0,2],[1,1,2],[0,2,2], [3,0,1],[2,1,1],[1,2,1], [0,3,1],[4,0,0],[3,1,0], [2,2,0],[1,3,0],[0,4,0]]; $triangulation = [[0,1,2],[9,11,12],[9,12,13], [9,13,14],[1,2,5],[6,10,11],[3,6,11],[1,5,9],[1,3,11], [8,9,11],[1,4,9],[1,7,11],[7,8,11],[4,8,9],[1,4,7],[4,7,8]]; $pointMatrix =(ones_vector<Rational>(15)) | (new Matrix<Rational>($points)); $Sigma = new SubdivisionOfPoints(POINTS=>$pointMatrix, MAXIMAL_CELLS=>$triangulation); $Sigma->VISUAL;

gives a beautiful subdivision. On the other hand, as mentioned in the same paper a few pages later, one specific lifting function on the 15 points which yields our example triangulation is [6, 0,3, 1,-1/3,1, 3,-1/3,-1/3,0, 6,0,1,3,6]; in other words, the subdivision of the same points with respect to this weight vector is again Sigma.

My question: How did the authors come up with this vector? Is there an algorithmic way (using a polymake function, for example)? Surely, I can compute such a vector by hand if I work hard enough, but in some cases subdivision in hand might be quite complicated.

In fancy words, I'd like to find an "actual point" in the secondary cone of Sigma:

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$sc=$Sigma->secondary_cone();

Thanks in advance!

[joswig]

Indeed, there are suitable functions. For instance,

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polytope > print $Sigma->secondary_cone()->REL_INT_POINT; 1/12 -7/12 0 -53/72 -19/36 0 -29/36 -61/72 -31/72 1/36 -19/24 -13/12 -7/12 0 2/3

gives you some such vector, produced via a linear program (finding an interior point in a polytope). No reason to expect this to be nice in any way.

Usually, this one is better:

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polytope > print $Sigma->MIN_WEIGHTS; 10 1 6 3 0 3 6 0 0 1 10 1 3 6 10

The above uses a certain integer linear program to find an integral vector with small coordinates.

[EnisKaya]

That's exactly what I need, thanks Michael!