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Record W4392712528 · doi:10.4153/s0008439524000201

Ideals with componentwise linear powers

2024· article· en· W4392712528 on OpenAlex

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

venuePublished in a venue whose home country is Canada.
no affNo Canadian affiliation: this work is invisible to an affiliation-only frame.
No Canadian affiliation. An affiliation-only frame, the usual design, would never have seen this work. It is one of the works that make the case for inverting the frame.

Bibliographic record

VenueCanadian Mathematical Bulletin · 2024
Typearticle
Languageen
FieldMathematics
TopicCommutative Algebra and Its Applications
Canadian institutionsnot available
FundersJapan Society for the Promotion of ScienceAlexander von Humboldt-Stiftung
KeywordsMathematicsPolynomial ringBipartite graphIdeal (ethics)CombinatoricsCover (algebra)Chordal graphQuadratic equationGraphDiscrete mathematicsPolynomialGeometryMathematical analysisLaw

Abstract

fetched live from OpenAlex

Abstract Let $S=K[x_1,\ldots ,x_n]$ be the polynomial ring over a field K , and let A be a finitely generated standard graded S -algebra. We show that if the defining ideal of A has a quadratic initial ideal, then all the graded components of A are componentwise linear. Applying this result to the Rees ring $\mathcal {R}(I)$ of a graded ideal I gives a criterion on I to have componentwise linear powers. Moreover, for any given graph G , a construction on G is presented which produces graphs whose cover ideals $I_G$ have componentwise linear powers. This, in particular, implies that for any Cohen–Macaulay Cameron–Walker graph G all powers of $I_G$ have linear resolutions. Moreover, forming a cone on special graphs like unmixed chordal graphs, path graphs, and Cohen–Macaulay bipartite graphs produces cover ideals with componentwise linear powers.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.

metaresearch head score (Codex)0.000
metaresearch head score (Gemma)0.000
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesInsufficient payload (model declined to judge)
Consensus categoriesInsufficient payload (model declined to judge)
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: none
Teacher disagreement score0.612
Threshold uncertainty score0.984

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0000.000
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0190.017

Machine scores (provisional)

The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.

Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.

Opus teacher head0.032
GPT teacher head0.299
Teacher spread0.267 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it