An Efficient Algorithm for Computing Parametric Multivariate Polynomial GCD
Bibliographic record
Abstract
A new efficient algorithm for computing a parametric greatest common divisor (GCD) of parametric multivariate polynomials over k[u][x] is presented. The algorithm is based on a well-known simple insight that the GCD of two multivariate polynomials (non-parametric as well as parametric) can be extracted using the generator of the quotient ideal of a polynomial with respect to the second polynomial. And, further, this generator can be obtained by computing a minimal Gröbner basis of the quotient ideal. The main attraction of this idea is that it generalizes to the parametric case for which a comprehensive Gröbner basis is constructed for the parametric quotient ideal. It is proved that in a minimal comprehensive Gröbner system of a parametric quotient ideal, each branch of specializations corresponds to a principal parametric ideal with a single generator. Using this generator, the parametric GCD of that branch is obtained by division. This algorithm does not need to consider whether parametric polynomials are primitive w.r.t. the main variable. This is in sharp contrast to two algorithms recently proposed by Nagasaka (ISSAC, 2017). The resulting algorithm is not only conceptually simple to understand but is considerably efficient. The proposed algorithm and both of Nagasaka's algorithms have been implemented in Singular (available at http://www.mmrc.iss.ac.cn/~dwang/software.html), and their performance is compared on a number of examples. For more than two polynomials, this process can be repeated by considering pairs of polynomials; the efficiency in that case becomes even more evident.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.012 | 0.006 |
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 source (direct Gemma or distilled Codex), 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".