Integer Complexity Generalizations in Various Rings

TL;DR

This paper explores generalizations of integer complexity across various algebraic structures, establishing bounds, introducing new concepts, and visualizing complexity to deepen understanding of these mathematical objects.

Contribution

It introduces new complexity measures and bounds for integers in cyclotomic rings, polynomials over naturals, and integers mod m, expanding the scope of integer complexity analysis.

Findings

Bounds established for cyclotomic rings

New concepts like inefficiency and resilience introduced

Graph visualizations of complexity in finite rings

Abstract

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.