Kazarnovskii Pseudovolume

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gino
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Joined: 26 Apr 2017, 10:59

Kazarnovskii Pseudovolume

Postby gino » 27 Apr 2017, 17:53

Good afternoon,

I would like to know if it is possible to use polymake in order to compute the Kazarnovskii pseudovolume of 4-dimensional polytopes.

If \( \Gamma \) is a polytope in \( \mathbb C^2 \), the Kazarnovskii pseudovolume \( P_2(\Gamma) \) is, by definition, the sum \( \frac{1}{\pi}\sum_\Delta \rho(\Delta)vol_2(\Delta)\psi(\Delta) \), as \( \Delta \) runs in the set of 2-dimensional faces of \( \Gamma \), where:
- \( \rho(\Delta)=1-\langle v_1,v_2\rangle^2 \), with \( \{v_1,v_2\} \) an orthonormal basis (respect to the scalar product \( Re\langle\,,\rangle \) given by the real part of the standard hermitian one) of the plane parallel to \( \Delta \) and passing through the origin;
- \( vol_2(\Delta) \) is the surface area of \( \Delta \);
- \( \psi(\Delta) \) is the outer angle of \( \Gamma \) at \( \Delta \).

So \( P_2(\Gamma) \) is just a weighted version of the 2nd intrinsic volume of \( \Gamma \) taking into account the position of \( \Gamma \) with respect to complex structure of the ambient space. My question is the following: is polymake able to perform the necessary linear algebra computation on the set of the ridges of \( \Gamma \)?

Thank you in advance

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joswig
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Joined: 24 Dec 2010, 11:10
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Re: Kazarnovskii Pseudovolume

Postby joswig » 28 Apr 2017, 17:05

This is a special computation which is not supported by polymake right away.

One thing which makes this a bit delicate is that this needs to be implemented with floats. By design polymake is primarily about exact computations. Therefore typical float linear algebra is dramatically underdeveloped in polymake. Essentially, the only non-trivial algorithm being singular value decomposition (and thus solving systems of linear equations with reasonable accuracy).

It seems doable though, by writing suitable C++ client code.

Just a general warning: starting out with float coordinates for your points or inequalities polymake (by default) will convert into exact rational numbers. The output can later be converted to floats; see, e.g., https://forum.polymake.org/viewtopic.php?f=9&t=7&p=24&hilit=float#p24.


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