Determination Problems for Orbit Closures and Matrix Groups

Computational problems concerning the orbit of a point under the action of a matrix group occur throughout computer science, including in program analysis, complexity theory, 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 a set of points and asks questions about the orbit closure of the set 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 matrix groups and orbit closures. These problems begin with a given variety and seek to understand whether and how it arises either as an algebraic matrix 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 and a number $s$ and determines whether the given variety arises as an orbit closure of a point under an $s$-generated commutative algebraic matrix group. The main tools in our approach are structural properties of commutative algebraic matrix groups and module theory. We leave open the question of determining whether a variety is an orbit closure of a point under an $s$-generated algebraic matrix group (without the requirement of commutativity).