Computed Exceptional Hilbert Series

Definitions and mathematical statement

For a field $k$ and positive integers $n,m,d$, let $S=k[x_1,\ldots,x_n]$ have its standard grading, with $\deg(x_i)=1$. Given linear forms $\ell_1,\ldots,\ell_m\in S_1$, define \[ R_{n,m,d}=S/(\ell_1^d,\ldots,\ell_m^d),\qquad H_{R_{n,m,d}}(t)=\sum_{j\geq 0}\dim_k(R_{n,m,d})_j\,t^j. \] These definitions explain the notation; they do not assert that the source's claimed behavior holds over every field or for every choice of linear forms. The source requires the linear forms to be general, but supplies neither hypotheses on $k$, including its characteristic, nor a precise meaning of generality.

The supplied assertion associates the following six parameter triples with monomials described as differences in leading terms from values predicted by Fröberg's conjecture: \[ \begin{array}{c|c} (n,m,d)&\text{associated monomial}\\ \hline (4,6,5)&10t^9\\ (6,8,3)&42t^6\\ (8,10,2)&15t^4\\ (8,10,3)&135t^8\\ (10,12,2)&22t^5\\ (10,12,3)&88t^{10} \end{array} \] The table preserves the source's ordering and coefficients. It does not specify an equality: the record does not determine whether these monomials are terms of the actual Hilbert series, terms of the predicted series, or leading terms of their difference. It also does not specify the direction of subtraction, the meaning of leading term, or the precise prediction used. Consequently, no uniquely determined Hilbert-series identity or quantified proposition can be recovered from this record.

The supplied verification note describes the report passage as a theorem listing computed exceptions to Fröberg's prediction, rather than an open problem. The registry nevertheless retains the label open and marks the statement unrecoverable. These are supplied curation claims and labels, not independently verified mathematical conclusions. No separate open question is specified.

Short English statement

The record lists six quotients by powers of general linear forms and six monomials associated with reported exceptions to Fröberg's prediction. Missing field hypotheses and an unclear comparison prevent a precise mathematical assertion. Supplied curation describes the passage as a theorem, although the registry retains an open label; human review of the source is required.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004530$, title: Computed Exceptional Hilbert Series; source code catalogue entry; collection: Oberwolfach Reports Open Problems. The record lists proposed year 2020 and names no proposer.
• Supplied report citation: Lefschetz Properties in Algebra, Geometry and Combinatorics (2021), Oberwolfach Reports, pp. 1539--1600, DOI 10.4171/owr/2020/31. Source URL and literature-source link labelled OWR report: https://doi.org/10.4171/owr/2020/31. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no exact location within these pages is supplied for this assertion.
• Supplied literature-source link labelled Stanley: https://doi.org/10.1016/0001-8708(78)90045-2. No title, full author attribution, or specific supporting passage is supplied.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the status open and describes the extracted text as a polynomial fragment lacking ring and genericity hypotheses, whose apparent comparison with Fröberg predictions cannot be classified as resolved or disproved. The background expressly requests independent verification of time-sensitive status and cited claims. The supplied record separately includes a verification note describing the report passage as a theorem of computed Hilbert-series exceptions and retaining the original record for provenance. Neither linked source has been independently consulted for this draft.