🤖 AI Summary
This work addresses the solvability of polynomial equations $AX^k = B$ in the semiring of finite discrete dynamical systems (FDDS), focusing on division operations and the existence, construction, and uniqueness of $k$-th roots for connected FDDS. Methodologically, it establishes the first iterative root theory for FDDS semirings, revealing algebraic correspondences between root structures and state-graph decomposition, cyclic classes, and homomorphic decompositions. Integrating semiring algebra, automata theory, graph theory, and category-theoretic methods, the paper introduces restricted homomorphisms and idempotent decomposition techniques. It fully characterizes necessary and sufficient conditions for an FDDS to admit an $n$-th root, provides a polynomial-time constructive algorithm, and proves uniqueness up to strongly connected components. These results establish a novel algebraic paradigm for modeling and controllability analysis of discrete dynamical systems.