发表机构
Durham University; Université de Lille, CNRS, Inria, UMR 9189 CRIStAL; Aix-Marseille Université, CNRS, LIS（杜伦大学; 里尔大学，法国国家科学研究中心，法国研究与数字创新署，CRIStAL联合实验室; 艾克斯-马赛大学，法国国家科学研究中心，LIS联合实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究部分动力系统（出度至多1的函数有向图）构成的交换半环，刻画其素元，并证明单射单变量多项式刻画与除法复杂度与完整动力系统半环一致。
AI 中文摘要
对可观测现象（例如在生物学或物理学中）的分析能够检测出动力学行为。如果条件理想且观测数量充足，我们可以用动力系统（也称为函数有向图，即每个节点的出度恰好为1的图）来表示这些现象。在同构意义下，这些动力系统以不相交并作为加法、直积作为乘法，构成一个交换半环。先前关于该半环的多项研究旨在建立代数性质（素性、单射性）或复杂性结果（除法、因式分解）。然而，尚未有工作针对由不完美观测（导致缺失转移或节点，即图中每个节点的出度至多为1）所得到的图展开研究。在这种情况下，我们称该系统为部分的。在本文中，我们证明了部分动力系统在同构意义下并配备相同的加法和乘法，仍然构成一个交换半环。接着，我们刻画了该半环的素元，这些素元与动力系统半环的素元不同。最后，我们强调了两个半环共有的两个性质。第一，单射单变量多项式在两者中具有相同的刻画。第二，对于部分动力系统，除法可以在多项式时间内计算当且仅当对于动力系统也可以。
英文摘要
The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviours. If the conditions are ideal and the number of observations is sufficient, we can represent these phenomena by a dynamical system, also called a functional digraph, that is to say a graph where each node has out-degree exactly one. Up to isomorphism, these dynamical systems, equipped with disjoint union as addition and direct product as multiplication, form a commutative semiring. Several previous studies on this semiring have aimed to establish algebraic properties (primality, injectivity) or complexity results (division, factorisation). However, no work has yet been conducted on graphs derived from imperfect observations that result in missing transitions or nodes, in other words, in cases where each node in the graph has an out-degree of at most one. In this case, we say that the system is partial. In this paper, we show that partial dynamical systems, up to isomorphism and equipped with the same addition and multiplication, still form a commutative semiring. We then characterise the prime elements of this semiring, which differ from those of the semiring of dynamical systems. Finally, we highlight two properties shared by both semirings. First, injective univariate polynomials admit the same characterisation in both. Second, division can be computed in polynomial time for partial dynamical systems if and only if it can be for dynamical systems.
