疎な多変数多項式系の高速な変数消去法の探求 (Computer Algebra : Theory and its Applications)
Bibliographic record
Abstract
Given {F_{1}, …, F_{m+1}}subset mathbb{K}[x, u]_{dot{t}} with mgeq 2, where mathbb{K} is a number field of characteristic 0_{:} (x) =(x_{1}, …, x_{m}) and (u) =(u_{1}, …, u_{n})_{dot{gamma}} we want to develop an efficient method of computing hat{S}(u) which is the smallest polynomial of the elimination ideal langle F_{1}, F_{m+1}ranglecap mathbb{K}[u]_{:} without computing the Gröbner basis but eliminating x with polynomial remainder sequences w.r. t. x. Our targets are sparse multivariate polynomials of many sub-variables u. Last year. we succeeded in finding such a method for {G, H}subset mathbb{K}[x, u]. The method is much more efficient than the Gröbner basis method when ngeq 3. In this paper, we attack the case of mgeq 2. Contrary to the conventional triangularization method which converts the initial system to {G_{1}(x_{1}, …, x_{m}, u), G_{2}(x_{2}, …, x_{m}, u), …, G_{m} (x_{m}, u), H(u)}_{dot{0}} our method computcs H_{1}(u), H_{l}(u). We will prove that each H_{i}(u)(iin{1, …, l}) is a multiple of hat{S}{dot{y} so we compute H :=gcd(H_{1}, …, H_{l}). H is a small multiple of hat{S}:H=overline{H}hat{S}. We propose a method of deleting "extraneous factor" overline{H}{backslash.} by factorizing H.
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How this classification was reachedexpand
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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".