Mahler’s work on Diophantine equations and subsequent developments
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Bibliographic record
Abstract
The main body of K. Mahler's work on Diophantine equations consists of his 1933 papers \[Math. Ann. 107, 691--730 (1933; Zbl 0006.10502; JFM 59.0220.01); 108, 37--55 (1933; Zbl 0006.15604); Acta Math. 62, 91--166 (1934; Zbl 0008.19801; JFM 60.0159.04)], in which he proved a generalization of the Thue-Siegel Theorem on the approximation of algebraic numbers by rationals, involving $p$-adic absolute values, and applied this to get finiteness results for the number of solutions for what became later known as Thue-Mahler equations. He was also the first to give upper bounds for the number of solutions of such equations. In fact, Mahler's extension of the Thue-Siegel Theorem made it possible to extend various finiteness results for Diophantine equations over the integers to $S$-integers, for any arbitrary finite set of primes $S$. For instance Mahler himself \[J. Reine Angew. Math. 170, 168--178 (1934; Zbl 0008.20002; JFM 60.0159.03)] extended Siegel's finiteness theorem on integral points on elliptic curves to $S$-integral points. In this chapter, we discuss Mahler's work on Diophantine approximation and its applications to Diophantine equations, in particular Thue-Mahler equations, $S$-unit equations and $S$-integral points on elliptic curves, and go into later developments concerning the number of solutions to Thue-Mahler equations and effective finiteness results for Thue-Mahler equations. For the latter we need estimates for $P$-adic logarithmic forms, which may be viewed as an outgrowth of Mahler's work on the $P$-adic Gel'fond-Schneider theorem \[Compos. Math. 2, 259--275 (1935; Zbl 0012.05302; JFM 61.0187.01)]. We also go briefly into decomposable form equations, these are certain higher dimensional generalizations of Thue-Mahler equations.
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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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.005 | 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 it