Representation of squares by monic second degree polynomials in the field of 𝑝-adic meromorphic functions
Bibliographic record
Abstract
We prove a result on the representation of squares by monic second degree polynomials in the field of p p -adic meromorphic functions in order to solve positively Büchi’s n n squares problem in this field. Using this result, we prove the non-existence of an algorithm to decide whether a system of diagonal quadratic forms over Z [ z ] \mathbb {Z}[z] represents or not in the ring of p p -adic entire functions (in the variable z z ) a given vector of polynomials in Z [ z ] \mathbb {Z}[z] , and a similar result for p p -adic meromorphic functions when the systems allow vanishing conditions on the unknowns. This improves the known negative answers for the analogue of Hilbert’s Tenth Problem for these structures. We also improve some results by Vojta concerning the case of complex meromorphic functions, the case of function fields and finally the case of number fields, and we show an intimate relation of the latter with Bombieri’s Conjecture for surfaces over number fields.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
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".