Finite Generation and Separation in Invariant Rings

Definitions and mathematical statement

Four invariant-theory questions. For a linear algebraic group $G$ acting on an affine variety $X$, is $K[X]^G$ finitely generated? If it is, find generators. Determine its structural properties, and determine whether invariant functions separate distinct $G$-orbits. This is a list of tasks over a broad class, not an asserted affirmative theorem for every group action.

Let $K$ be the intended field, $A=K[X]$, and suppose the action is algebraic, with coaction $\rho:A\to K[G]\otimes_KA$. The invariant ring is $A^G=\{f\in A:\rho(f)=1\otimes f\}$. Over an algebraically closed field this is equivalently the ring of regular functions satisfying $f(gx)=f(x)$ for all $g,x$. Finite generation asks for finitely many $f_1,\ldots,f_r\in A^G$ with $A^G=K[f_1,\ldots,f_r]$; the computational question asks to determine such a list from a specified presentation of the action when possible. No reductivity, characteristic-zero, or finite-group hypothesis is supplied. Those distinctions must be retained when determining the scope of an answer.

Separation of geometric orbits means that for every pair $x,y$ in different $G$-orbits over the chosen algebraic closure, some invariant regular function has $f(x)\ne f(y)$. The source does not specify geometric versus $K$-rational orbits or whether only closed orbits are intended. Regular invariants are constant on an orbit closure, so intersecting orbit closures cannot be separated by them; separating distinct closed orbits is therefore a different request from separating every pair of distinct orbits. The ring's unnamed structural properties and the algorithmic input or complexity expectations are also unspecified. No particular structure theorem is inserted.

The supplied triage checked 2026-08-21 labels the record open but explicitly says it is a broad list with general cases rather than one claim. It supplies no class-by-class theorem or resolved-status statement. That research-program qualification is retained, and the open label is not used to imply that every basic subquestion is unsolved or has an affirmative answer.

Short English statement

Study finite generation, explicit generators, structure, and orbit separation for the specified invariant ring. The field and group-action class matter, and separation of all orbits must be distinguished from separation of closed orbits. The source presents a program of questions rather than one universal conjecture.

Sources

• Local UnsolvedMath registry, permanent ID 30000524, code catalogue entry; Finite Generation and Separation in Invariant Rings, Oberwolfach Reports Open Problems.
• Computational Group Theory (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/30, pp. 1795--1878. https://doi.org/10.4171/owr/2006/30. Proposed year and year tag: 2006; no proposer named.
• Supplied literature label original Oberwolfach report: https://doi.org/10.4171/owr/2006/30. No separate authoritative finite-generation, algorithmic, or separation theorem citation is supplied.
• Literature triage checked 2026-08-21, status open with a broad-question-list qualification; record updated 2026-09-16. The corrected_verified field concerns delimiters and preservation of all subquestions. Verify dated claims independently.