Computing Independent Variable Sets for Polynomial 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
Computing independent variable sets for polynomial ideals plays an important role in solving high‐dimensional polynomial equations. The computation of a Gröbner basis for an ideal, with respect to a block lexicographical order in classic methods, is huge, and then an improved algorithm is given. Based on the quasi‐Gröbner basis of the extended ideal, a criterion of assigning independent variables is gained. According to the criteria, a maximal independent variable set for a polynomial ideal can be computed by assigning indeterminates gradually. The key point of the algorithm is to reduce dimensions so that the unit of computation is one variable instead of a set, which turns a multivariate problem into a single‐variable problem and turns the computation of rational function field into that of the fundamental number field. Hence, the computation complexity is reduced. The algorithm has been analysed by an example, and the results reveal that the algorithm is correct and effective.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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 | 0.000 |
| 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