Question: We say that a convex lattice polytope $P\subset \mathbb{R}^d$ is a smooth Fano polytope if:

  1. The origin is contained in the interior of $P$
  2. The vertices of every facet of $P$ are a $\mathbb{Z}$-basis of $\mathbb{Z}^d$

Now, suppose we have a set $V=\{v_1,\ldots,v_n\}\subset \mathbb{Z}^d$. What is an (preferentially efficient) algorithm for deciding if the convex hull $\text{conv}(V)$ is a smooth Fano polytope?

I've read a paper that gives an algorithm to classify all smooth Fano polytopes given the dimension $d$ as input (An algorithm for the classification of smooth Fano polytopes by Mikkel Øbro), and while trying to implement said algorithm I discovered I don't know how to solve this question.
For those who are curious, I want to use the algorithm to help me gain some intuition about toric varieties, and use it to compute known invariants and test conjectures.

  • $\begingroup$ It's straightforward to find an algorithm to find the facets of P, which by #2 would each have to have just d vertices; thus P must be simplicial. The d vertices, subtending a facet, could not fail to be linearly independent. To show they form a $\mathbb{Z}$-basis, would it suffice to show that the GCD of each of the $x_i$ components of the vertices is 1 for i = 1 to d? $\endgroup$ – Dan Moore Jan 18 '11 at 22:42
  • $\begingroup$ @DanMoore This is not sufficient. If you take, say (1,2) and (2,1) the gcds are 1, but the determinant is 3, so they cannot cover everything. $\endgroup$ – Phira Nov 2 '11 at 18:25
  • $\begingroup$ What's wrong with Gaussian elimination? $\endgroup$ – Stephen Feb 29 '12 at 15:50

After you have found the facets of $P$, you can check if $d$ points form a $\mathbb Z$ bases by calculating their determinant which has to be $\pm 1$.

(The determinant is the volume of the fundamental domain and is the inverse of the density of lattice points. A sublattice is the integer lattice if and only if the density is the same.)


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