We study the computational complexity of decomposing finite discrete dynamical systems (FDDSs) in terms of the semiring operations of alternative and synchronous execution, which is useful for the analysis of discrete phenomena in science and engineering. More specifically, we investigate univariate polynomials of the form $P(X) = B$, that is with a constant side, first over the subsemiring of permutations and then over general FDDSs. We find a characterization of injective polynomials $P$ and efficient algorithms for solving the associated equations. Then, we introduce the more general notion of pseudo-injective polynomial, which is based on a condition on the lengths of the limit cycles of its coefficients, and prove that the corresponding equations are also solvable efficiently. These results also apply even when permutations are encoded in an exponentially more compact way.

翻译：我们研究了有限离散动力系统（FDDSs）在交替执行与同步执行的半环运算下分解的计算复杂度，这对科学与工程中离散现象的分析具有重要价值。具体而言，我们考察了形如 $P(X) = B$（即具有常数项）的一元多项式，首先在置换的子半环上，随后在一般FDDSs上展开研究。我们给出了单射多项式 $P$ 的表征，并提出了求解相关方程的高效算法。接着，我们引入了更广义的伪单射多项式概念，该概念基于其系数极限环长度的条件，并证明了相应方程同样可高效求解。上述结果在置换以指数级更紧凑方式编码时依然成立。