Computational problems concerning the orbit of a point under the action of a matrix group occur in numerous subfields of computer science, including complexity theory, program analysis, quantum computation, and automata theory. In many cases the focus extends beyond orbits proper to orbit closures under a suitable topology. Typically one starts from a group and several points and asks questions about the orbit closure of the points under the action of the group, e.g., whether two given orbit closures intersect. In this paper we consider a collection of what we call determination problems concerning groups and orbit closures. These problems begin with a given variety and seek to understand whether and how it arises either as an algebraic group or as an orbit closure. The how question asks whether the underlying group is $s$-generated, meaning it is topologically generated by $s$ matrices for a given number $s$. Among other applications, problems of this type have recently been studied in the context of synthesising loops subject to certain specified invariants on program variables. Our main result is a polynomial-space procedure that inputs a variety $V$ and a number $s$ and determines whether $V$ arises as an orbit closure of a point under an $s$-generated commutative matrix group. The main tools in our approach are rooted in structural properties of commutative algebraic matrix groups and lattice theory. We leave open the question of determining whether a variety is an orbit closure of a point under an algebraic matrix group (without the requirement of commutativity). In this regard, we note that a recent paper by Nosan et al. [NPSHW2021] gives an elementary procedure to compute the orbit closure of a point under finitely many matrices.

翻译：关于矩阵群作用下点的轨道的计算问题出现在计算机科学的众多子领域中，包括复杂性理论、程序分析、量子计算和自动机理论。在许多情况下，研究重点从轨道本身扩展到适当拓扑下的轨道闭包。通常从群和若干点出发，研究这些点在群作用下的轨道闭包问题，例如判断两个给定轨道闭包是否相交。本文研究一类关于群与轨道闭包的判定问题。这类问题从给定簇出发，试图理解其是否以及如何作为代数群或轨道闭包出现。其中“如何出现”问题探究底层群是否为$s$生成，即在给定数$s$下该群是否可由$s$个矩阵拓扑生成。除其他应用外，此类问题近期在程序变量满足特定不变式的循环综合研究中受到关注。我们的主要结果是多项式空间算法：输入簇$V$和数$s$，判定$V$是否可作为某$s$生成交换矩阵群作用下点的轨道闭包出现。方法的核心工具源于交换代数矩阵群的结构性质与格理论。我们留下未解决问题：如何判定簇是否为代数矩阵群作用下点的轨道闭包（不要求交换性）。对此我们注意到，Nosan等人近期论文[NPSHW2021]给出了有限个矩阵作用下点轨道闭包的基本计算方法。