Newton–Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals
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.
Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper I equals left-brace upper I Subscript k Baseline right-brace Subscript k element-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">I</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {I}=\{I_k\}_{k \in \mathbb {N}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a graded family of monomial ideals. We use the Newton–Okounkov body of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper I"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {I}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to: (a) give a characterization for the Noetherian property of the Rees algebra of the family <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper I"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {I}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; and (b) present a combinatorial interpretation for the analytic spread of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper I"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {I}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also apply these results to investigate and give bounds for the generation type and the Veronese degree of the symbolic Rees algebra of a monomial ideal.
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 imitationNot 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.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.005 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it