Simultaneously preperiodic points for a family of polynomials in positive characteristic
Bibliographic record
Abstract
Abstract In the goundbreaking paper [BD11] (which opened a wide avenue of research regarding unlikely intersections in arithmetic dynamics), Baker and DeMarco prove that for the family of polynomials $f_\lambda (x):=x^d+\lambda $ (parameterized by $\lambda \in \mathbb {C}$ ), given two starting points a and b in $\mathbb {C}$ , if there exist infinitely many $\lambda \in \mathbb {C}$ such that both a and b are preperiodic under the action of $f_\lambda $ , then $a^d=b^d$ . In this paper, we study the same question, this time working in a field of characteristic $p>0$ . The answer in positive characteristic is more nuanced, as there are three distinct cases: (i) both starting points a and b live in ${\overline {\mathbb F}_p}$ ; (ii) d is a power of p ; and (iii) not both a and b live in ${\overline {\mathbb F}_p}$ , while d is not a power of p . Only in case (iii), one derives the same conclusion as in characteristic $0$ (i.e., that $a^d=b^d$ ). In case (i), one has that for each $\lambda \in {\overline {\mathbb F}_p}$ , both a and b are preperiodic under the action of $f_\lambda $ , while in case (ii), one obtains that also whenever $a-b\in {\overline {\mathbb F}_p}$ , then for each parameter $\lambda $ , we have that a is preperiodic under the action of $f_\lambda $ if and only if b is preperiodic under the action of $f_\lambda $ .
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.004 | 0.004 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 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".