Determination Problems for Orbit Closures and Matrix Groups
Rida Ait El Manssour, George Kenison, Mahsa Shirmohammadi, Anton Varonka, James Worrell
TL;DR
The paper addresses the problem of deciding when a given algebraic variety is an orbit closure under an $s$-generated commutative algebraic matrix group, and when such a variety is the orbit closure of a vector under the action of a group with at most $s$ generators. It develops a polynomial-space decision procedure by reducing to fragments of the first-order theory of algebraically closed or real-closed fields, and it leverages a structural decomposition into semisimple and unipotent parts, plus a base-change to canonical forms and lattice-binomial descriptions, to handle both orbit-closure and group-determination questions. The results include detailed procedures for commuting matrices, a clear separation of diagonal and unipotent components, and constructive algorithms to compute generators in the cyclic-case, with applications to loop synthesis and program analysis. Overall, the work advances understanding of when polynomial invariants and orbit closures arise from finitely generated linear groups and provides efficient (polynomial-space) methods for these determination problems in the commutative setting.
Abstract
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).