New Algorithms for Exact and Approximate Polynomial Decomposition
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Bibliographic record
Abstract
Computing a decomposition of a polynomial f (x) as a functional composition g(h(x)) of polynomials g(x) and h(x) is an important and well-studied problem, both for exact and approximate inputs. In this paper, we re-examine the original (exponential-time) algorithm of Barton and Zippel for this task, which looks for special factors of an associated separated bivariate polynomial. We demonstrate algorithms using this approach which are reasonably fast (i.e., run in a polynomial number of operations) for exact computation, and provide an effective new approach for the decomposition of approximate polynomials. For approximate polynomials we exhibit rigorous lower bounds on the distance to the nearest decomposable polynomial, as well as robust numerical algorithms.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it