THE NON-CYCLOTOMIC PART OF f(x)x<sup>n</sup>+ g(x) AND ROOTS OF RECIPROCAL POLYNOMIALS OFF THE UNIT CIRCLE
Bibliographic record
Abstract
Given relatively prime polynomials f(x) and g(x) in ℤ[x] with non-zero constant terms, we show that for n greater than an explicitly determined bound depending on f(x) and g(x), if the polynomial f(x)x n + g(x) is non-reciprocal, then its non-cyclotomic part is irreducible except for some explicit cases where a known factorization of f(x)x n + g(x) can easily be described. Prior work of a similar nature is discussed which shows under similar circumstances the non-reciprocal part off(x)x n + g(x) is irreducible. The current paper establishes and makes use of a result which shows that a reciprocal polynomial f(x) with a root off the unit circle must have a root bounded away from the unit circle by an explicitly given function of the degree of f(x), the leading coefficient a of f(x) and the discriminant of f(x). Notably in this result, a need not be 1.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.003 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 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".