  Cohomological ideas have recently been injected into persistent homology and
have for example been used for accelerating the calculation of persistence
diagrams by the software Ripser.
  The cup product operation which is available at cohomology level gives rise
to a graded ring structure that extends the usual vector space structure and is
therefore able to extract and encode additional rich information. The maximum
number of cocycles having non-zero cup product yields an invariant, the
cup-length, which is useful for discriminating spaces.
  In this paper, we lift the cup-length into the persistent cup-length function
for the purpose of capturing ring-theoretic information about the evolution of
the cohomology (ring) structure across a filtration. We show that the
persistent cup-length function can be computed from a family of representative
cocycles and devise a polynomial time algorithm for its computation. We
furthermore show that this invariant is stable under suitable interleaving-type
distances.
