Asymptotic Resurgence of facet ideals of matroids in Sage
Load functions to compute skew Waldschmidt constant and asymptotic resurgence
Click the "Evaluate" button below to define the functions in the file Asymptotic resurgence of facet ideal of matroid .
Uniform matroids
It is well-known that the Waldschmidt constant of the facet ideal of the uniform matroid \(U_{k,n}\) of rank \(k\) on \(n\) elements is \(\frac{n}{n-k+1}\) and its asymptotic resurgence is \(\frac{k(n-k+1)}{n}\).
Almost uniform matroids
An almost uniform matroid \(U_{k,n}^-\) of rank \(k\) on a groundset of size \(n\) is a matroid whose bases are all the bases of the uniform matroid \(U_{k,n}\) except for one. We prove in Asymptotic Resurgence of facet and Stanley-Reisner ideals of matroids that the Waldschmidt constant of the facet ideal of \(U_{k,n}^-\) is \(\frac{n-1}{n-k}\) if \(2k\le n+1\) and \(\frac{n+1}{n-k+1}\) if \(2k\ge n+1\). Similarly, the asymptotic resurgence of the facet ideal of \(U_{k,n}^-\) is \(\frac{k(n-k)}{n-1}\) if \(2k\le n+1\) and \(\frac{(k-1)(n-k+1)}{n-1}\) if \(2k\ge n+1\).
The asymptotic resurgence is achieved in the direction of the complement of the empty flat. That is, it is given by the rank of the matroid divided by the Waldschmidt constant.
The asymptotic resurgence is achieved in the direction of the complement of a flat whose complement contains the missing basis 'cdef'.
Checking asymptotic resurgence for all non-isomorphic matroids of small rank
We can verify the data in Table 1 of Asymptotic Resurgence of facet and Stanley-Reisner ideals of matroids with variations on the following code (this has to be run on a local sage installation, as the matroids database is not loaded in the online version). For instance, the following code will verify that three of the four non-isomorphic simple matroids of rank 3 on 5 elements have facet ideals whose asymptotic resurgence is given by the rank divided by the Waldshmidt constant.