Double Jordan Stratifications of Artinian Gorenstein Algebras

Definitions and mathematical statement

In the usual commutative convention, an algebra $A$ is Artinian if its ideals satisfy the descending chain condition. It is Gorenstein if, for every maximal ideal $\mathfrak m$, the socle \[ \operatorname{Soc}(A_{\mathfrak m})=\{z\in A_{\mathfrak m}:\mathfrak m A_{\mathfrak m}z=0\} \] is one-dimensional over the residue field $A_{\mathfrak m}/\mathfrak m A_{\mathfrak m}$. An ideal is principal if it has the form $(u)=Au$. The source does not specify a base field, its characteristic, whether the algebra is local, or the grading assumptions for all parts of the question.

Requested constructions. In the intended class of Artinian Gorenstein algebras, the supplied record asks the following questions.

1. Given principal ideals $J=(u)$ and $K=(v)$, can the double $Q_{J,K}$-stratification be redefined so that its pieces have the expected dimensions and it retains its relation to the single-ideal stratifications $Q_J(a)$ and $Q_K(a)$?
2. For nongraded Artinian Gorenstein algebras, can the $Q_{\ell,m}$-decomposition be used to define a Jordan order type?
3. For a binomial, can the $Q_{x,y}$-decomposition be determined by combinatorial invariants of its associated poset? Can such a description also cover variables of different degrees and the case of three variables?

Unresolved formulation. The supplied record does not define any of these $Q$-stratifications or decompositions, the index $a$, the expected dimensions, or the relations that a redefinition must preserve. It also does not define Jordan order type, specify the roles or admissible choices of $\ell,m,x,y$, or construct an algebra and an associated poset from a binomial. Thus these are source-attributed construction questions; the supplied information does not determine a precise existence assertion or a combinatorial determination theorem. In particular, no choice of grading, binomial construction, poset, or dimension formula is imposed here.

Short English statement

The source asks for compatible double stratifications for two principal ideals, a Jordan order type for nongraded Artinian Gorenstein algebras, and a combinatorial description of the binomial case, including unequal variable degrees and three variables. The defining constructions and target conditions are missing, so a fully precise mathematical statement requires further source information.

Sources

• Local UnsolvedMath registry: permanent ID $30004525$, title Double Jordan Stratifications of Artinian Gorenstein Algebras, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2020; no individual proposer is identified.
• Supplied report citation: Lefschetz Properties in Algebra, Geometry and Combinatorics (2021), Oberwolfach Reports, pp. 1539--1600, DOI 10.4171/owr/2020/31. https://doi.org/10.4171/owr/2020/31. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The literature-source list labels this reference Oberwolfach Reports 2020/31.
• Supplied literature reference: Iarrobino--Macias Marques--McDaniel, Jordan Type. https://arxiv.org/abs/1811.02072.
• The supplied literature assessment, dated exactly 2026-08-22, labels the questions open and states that the proposed generalizations and combinatorial descriptions remain open, while existing work establishes only special Jordan-string decompositions. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry supplies the statement-status label corrected_verified and says that mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired for self-contained publication. This is a supplied editorial assertion; the record reproduced here still omits definitions needed for a precise formulation.