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Bibliographic record
Abstract
Abstract We consider a Weitzenböck derivation Δ acting on a polynomial ring <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>R</m:mi> <m:mo>=</m:mo> <m:mi>K</m:mi> <m:mo>[</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:mo>...</m:mo> <m:mo>,</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo>]</m:mo> </m:mrow> </m:math> $ R=K[\xi _1,\xi _2,\ldots ,\xi _m] $ over a field K of characteristic 0. The K -algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>R</m:mi> <m:mi>Δ</m:mi> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mi>h</m:mi> <m:mo>∈</m:mo> <m:mi>R</m:mi> <m:mo>∣</m:mo> <m:mi>Δ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>h</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>}</m:mo> </m:mrow> </m:mrow> </m:math> ${R^\Delta = \lbrace h \in R \mid \Delta (h) = 0\rbrace }$ is called the algebra of constants. Nowicki considered the case where the Jordan matrix for Δ acting on R 1 , the degree 1 component of R , has only Jordan blocks of size 2. He conjectured that a certain set generates <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>R</m:mi> <m:mi>Δ</m:mi> </m:msup> </m:math> $R^{\Delta }$ in that case. Recently Khoury, Drensky and Makar-Limanov and Kuroda have given proofs of Nowicki's conjecture. Here we consider the case where the Jordan matrix for Δ acting on R 1 has only Jordan blocks of size at most 3. We use combinatorial methods to give a minimal set of generators 𝒢 for the algebra of constants <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>R</m:mi> <m:mi>Δ</m:mi> </m:msup> </m:math> $R^{\Delta }$ . Moreover, we show how our proof yields an algorithm to express any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>h</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>R</m:mi> <m:mi>Δ</m:mi> </m:msup> </m:mrow> </m:math> $h \in R^\Delta $ as a polynomial in the elements of 𝒢. In particular, our solution shows how the classical techniques of polarization and restitution may be used to augment the techniques of SAGBI bases to construct generating sets for subalgebras.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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