Bibliographic record
Abstract
We present a new algorithm for determining all factorizations of a polynomial f in the domain ZN[x], a non-unique factorization domain, given in terms of parameters. From the prime factorization of N, the problem is reduced to factorization in Zph[x] where p is a prime and k ⪈ 1. If pk does not divide the discriminant of f and one factorization is given, our algorithm determines all factorizations with complexity Ο(n3 M(k log p)) where n denotes the degree of the input polynomial and M(t) denotes the complexity of multiplication of two t-bit numbers. Our algorithm improves on the method of von zur Gathen and Hartlieb, which has complexity Ο(n7 k(klog p + log n2). The improvement is achieved by processing all factors at the same time instead of one at a time and by computing the kernels and determinants of matrices over Zpk in an efficient manner.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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 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".