Canonical Heights and Preperiodic Points for Certain Weighted Homogeneous Families of Polynomials
Bibliographic record
Abstract
Abstract A family f of polynomials over a number field K will be called weighted homogeneous if and only if ft(z) = F(ze, t) for some binary homogeneous form F(X, Y) and some integer e ≥ 2. For example, the family zd + t is weighted homogeneous. We prove a lower bound on the canonical height, of the form \begin{align*} \hat{h}_{f_{t}}(z)\geq \varepsilon \max\!\left\{h_{\mathsf{M}_{d}}(f_{t}), \log|\operatorname{Norm}\mathfrak{R}_{f_{t}}|\right\},\end{align*} for values z ∈ K which are not preperiodic for ft. Here ε depends only on the number field K, the family f, and the number of places at which ft has bad reduction. For suitably generic morphisms $\varphi :\mathbb {P}^{1}\to \mathbb {P}^{1}$, we also prove an absolute bound of this form for t in the image of φ over K (assuming the abc Conjecture), as well as uniform bounds on the number of preperiodic points (unconditionally).
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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 | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 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) | 0.000 | 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 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".