Characterization of Hermite Ideal Projectors

Definitions and mathematical statement

The supplied question is: "Which finite-dimensional ideal projectors are Hermite projectors?"

A conditional polynomial-ring framework makes part of this question precise. For a specified field $K$ and positive integer $d$, put $A=K[x_1,\ldots,x_d]$. A finite-rank ideal projector on $A$ is a $K$-linear map $P:A\to A$ satisfying \[ P^2=P,\qquad \dim_K\operatorname{im}P<\infty, \] whose kernel is an ideal; equivalently, \[ \forall f\in\ker P\;\forall g\in A,\qquad gf\in\ker P. \] Here finite-dimensionality refers to the image of the projector.

In this framework, the requested classification would give necessary and sufficient conditions for such a projector $P$ to be a Hermite projector. However, the supplied record does not specify the underlying field, the number of variables, or a definition of "Hermite projector." In particular, it supplies no evaluation or derivative conditions defining that class. The polynomial-ring framework above is therefore conditional, and does not settle the intended domain or meaning of the source question. A definitive quantified classification problem requires these missing specifications; no particular definition of Hermite projector is imposed in this draft.

Short English statement

The question asks which finite-dimensional ideal projectors belong to the class of Hermite projectors. The supplied record does not define that class or specify the underlying polynomial setting, so the precise intended classification remains unresolved.

Sources

• Local registry provenance: the supplied UnsolvedMath record with permanent ID $30002864$, source problem code catalogue entry, and title "Characterization of Hermite Ideal Projectors." The project identifies data/unsolvedmath/problems.json and catalog/problems.json as registry files; their contents were not independently inspected for this draft. The record attributes the question to the Oberwolfach Reports collection and identifies no individual proposer.
• Supplied report citation: Multivariate Splines and Algebraic Geometry (2016), Oberwolfach Reports, pp. 1139--1200, DOI 10.4171/owr/2015/21. https://doi.org/10.4171/owr/2015/21. The record gives proposed year 2015 and describes the reports as documenting mathematical workshops and their open-problem discussions. The relation between the supplied 2016 title date and the 2015 provenance remains unverified.
• Supplied literature reference: Shekhtman, On One Class of Hermite Projectors. https://doi.org/10.1007/s00365-015-9311-5.
• Supplied literature reference: Shekhtman--Tuesink, Symmetry 2023. https://doi.org/10.3390/sym15091658.
• The exact supplied literature-check date is 2026-08-21. This is supplied dated curation, not an independently verified assessment. It labels the problem open and reports substantial structural results for finite-dimensional Hermite projectors, including complete-intersection and primary-decomposition criteria, while reporting no general characterization of all finite-dimensional ideal projectors that are Hermite projectors. The record expressly calls this a dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.