We present two methods to algorithmically compute both least and greatest solutions of polynomial equation systems over absorptive semirings (with certain completeness and continuity assumptions), such as the tropical semiring. Both methods require a polynomial number of semiring operations, including semiring addition, multiplication and an infinitary power operation. Our main result is a closed-form solution for least and greatest fixed points based on the fixed-point iteration. The proof builds on the notion of (possibly infinite) derivation trees; a careful analysis of the shape of these trees allows us to collapse the fixed-point iteration to a linear number of steps. The second method is an iterative symbolic computation in the semiring of generalized absorptive polynomials, largely based on results on Kleene algebras.

翻译：我们提出了两种方法,用算法来计算对吸收半径(具有一定完整性和连续性假设)(如热带半径)的多元等式系统最起码和最大的解决办法,例如热带半径。两种方法都需要半径作业的多元数,包括半环加法、倍增法和无限功率操作。我们的主要结果是基于固定点迭代法的最小和最大固定点的封闭式解决办法。证据基于(可能无限的)衍生树的概念;对这些树的形状进行仔细分析,使我们得以将固定点迭代法折成一系列线性步骤。第二种方法是在普遍吸收多波段半径的半径中进行迭代符号计算,主要基于Kleene代数的结果。