In this paper, we investigate generalizations of the Mahler-Popkens
complexity of integers. Specifically, we generalize to k-th roots of unity,
polynomials over the naturals, and the integers mod m. In cyclotomic rings,
we establish upper and lower bounds for integer complexity, investigate the
complexity of roots of unity using cyclotomic polynomials, and introduce a
concept of "minimality.'' In polynomials over the naturals, we establish bounds
on the sizes of complexity classes and establish a trivial but useful upper
bound. In the integers mod m, we introduce the concepts of "inefficiency'',
"resilience'', and "modified complexity.'' In hopes of improving the upper
bound on the complexity of the most complex element mod m, we also use graphs
to visualize complexity in these finite rings.Comment: 44 pages, 11 figures, Research Lab from PROMY