Bibliographic record
Abstract
We present three algorithms to calculate Φ n ( z ) \Phi _n(z) , the n t h n_{th} cyclotomic polynomial. The first algorithm calculates Φ n ( z ) \Phi _n(z) by a series of polynomial divisions, which we perform using the fast Fourier transform. The second algorithm calculates Φ n ( z ) \Phi _n(z) as a quotient of products of sparse power series. These two algorithms, described in detail in the paper, were used to calculate cyclotomic polynomials of large height and length. In particular, we have found the least n n for which the height of Φ n ( z ) \Phi _n(z) is greater than n n , n 2 n^2 , n 3 n^3 , and n 4 n^4 , respectively. The third algorithm, the big prime algorithm, generates the terms of Φ n ( z ) \Phi _n(z) sequentially, in a manner which reduces the memory cost. We use the big prime algorithm to find the minimal known height of cyclotomic polynomials of order five. We include these results as well as other examples of cyclotomic polynomials of unusually large height, and bounds on the coefficient of the term of degree k k
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.066 | 0.024 |
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".