论文标题
各种环的整数复杂性概括
Integer Complexity Generalizations in Various Rings
论文作者
论文摘要
在本文中，我们研究了整数的Mahler-Popkens复杂性的概括。具体而言，我们将统一的根源，多项式概括为$ k $ th，而整数$ m $。在环形环中，我们建立了整数复杂性的上限和下限，研究了使用循环组合多项式的统一根的复杂性，并引入了一个“最小值”的概念。在多项式上的自然元素中，我们建立了复杂性的尺寸，并建立了一个琐碎的上限。 “弹性”和“修改的复杂性。”，希望改善对最复杂的元素mod $ m $复杂性的上限，我们还使用图形来可视化这些有限环中的复杂性。
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.
