On the functional decomposition of multivariate laurent polynomials (abstract only)
Bibliographic record
Abstract
Determining whether a univariate polynomial may be written as the functional composition of two others of lower degree is a question that has been studied since at least the work of Ritt [1]. Algorithms by Barton and Zippel [2] and then by Kozen and Landau [3] have been incorporated in many computer algebra systems. Generalizations have been studied for functional decomposition of rational functions [4], algebraic functions [5], multivariate polynomials [6] and univariate Laurent polynomials [7]. We explore the functional decomposition problem for multivariate Laurent polynomials, considering the case f = g o h where g is univariate and h may be multivariate. We present an algorithm to find such a decomposition if it exists. The algorithm proceeds as follows: First, a variable weighting is chosen to make the weighted degree zero term in f constant. The positive degree and negative degree parts of h are then reconstructed separately, in a manner similar to that of Kozen and Landau, but by treating the homogeneous collections of terms by grade rather than individual monomials. Then terms of the univariate polynomial g are reconstructed degree by degree using a generic univariate projection of h.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".