Finite discrete-time dynamical systems (FDDS) model phenomena that evolve deterministically in discrete time. It is possible to define sum and product operations on these systems (disjoint union and direct product, respectively) giving a commutative semiring. This algebraic structure led to several works employing polynomial equations to model hypotheses on phenomena modelled using FDDS. To solve these equations, algorithms for performing the division and computing $k$-th roots are needed. In this paper, we propose two polynomial algorithms for these tasks, under the condition that the result is a connected FDDS. This ultimately leads to an efficient solution to equations of the type $AX^k=B$ for connected $X$. These results are some of the important final steps for solving more general polynomial equations on FDDS.

翻译：有限离散时间动力系统（FDDS）可用于模拟在离散时间内确定性演化的现象。可在这些系统上定义加法与乘法运算（分别为不交并和直积），从而构成一个交换半环。这一代数结构催生了多项工作，通过多项式方程对使用FDDS建模的现象假设进行形式化。为求解这些方程，需要实现除法运算及计算k次根的算法。本文针对结果必须是连通FDDS的条件，提出两种多项式时间算法，分别用于完成上述任务。这最终为求解连通X的方程AX^k=B提供了高效解法。这些成果是解决FDDS上更一般多项式方程的重要最终步骤之一。