ON THE EQUATION <i>f</i>(<i>g</i>(<i>x</i>))=<i>f</i>(<i>x</i>)<i>h</i><sup><i>m</i></sup>(<i>x</i>) FOR COMPOSITE POLYNOMIALS
Bibliographic record
Abstract
Abstract In this paper we solve the equation f ( g ( x ))= f ( x ) h m ( x ) where f ( x ), g ( x ) and h ( x ) are unknown polynomials with coefficients in an arbitrary field K , f ( x ) is nonconstant and separable, deg g ≥2, the polynomial g ( x ) has nonzero derivative g′ ( x )≠0 in K [ x ] and the integer m ≥2 is not divisible by the characteristic of the field K . We prove that this equation has no solutions if deg f ≥3 . If deg f =2 , we prove that m =2 and give all solutions explicitly in terms of Chebyshev polynomials. The Diophantine applications for such polynomials f ( x ) , g ( x ) , h ( x ) with coefficients in ℚ or ℤ are considered in the context of the conjecture of Cassaigne et al . on the values of Liouville’s λ function at points f ( r ) , r ∈ℚ.
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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 | 0.010 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".