Is it likely that in the future, there will be interest in computing persistent homology over the integers (or other PIDs)?


Currently, persistent homology is usually done over a field (like $\mathbb{Z}/2$), as the algorithms for producing the barcode only work for a field.


However, working over a field loses a lot of information. Also, there are algorithms that can compute the persistent groups over a PID (<http://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf> Section 5).


Thanks for any help. References will be very welcome. 



Persistent homology over the integers
As mentioned in Carlsson and Zomorodian's paper (to which you have linked), the problem of computing persistence barcodes with coefficients in a ring $R$ relies essentially on classifying graded modules over the polynomial ring $R[t]$. If (and only if!) $R$ is a field, $R[t]$ is a principal ideal domain and isomorphism classes of modules over it are easy to describe. When $R = \mathbb{Z}$, the modules are well-known to be hideous. Similar problems have plagued multi-dimensional persistence, since $R[s,t]$-modules are also pretty hard to classify (even when $R$ is a field).


But if you're willing to invest a bit in the Grothendieck construction, there might be hope: <https://arxiv.org/abs/1601.03107>


