  In 2012, Diem introduced a new figure of merit for cryptographic sequences
called expansion complexity. Recently, a series of paper has been published for
analysis of expansion complexity and for testing sequences in terms of this new
measure of randomness. In this paper, we continue this analysis. First we study
the expansion complexity in terms of the Gr\"obner basis of the underlying
polynomial ideal. Next, we prove bounds on the expansion complexity for random
sequences. Finally, we study the expansion complexity of sequences defined by
differential equations, including the inversive generator.
