Hilbert’s 13th problem in prime characteristic
Why this work is in the frame
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Bibliographic record
Abstract
The resolvent degree \operatorname{rd}_{\mathbb{C}}(n) is the smallest integer d such that a root of the general polynomial f(x) = x^{n} + a_{1} x^{n-1}+ \dots + a_{n} can be expressed as a composition of algebraic functions in at most d variables with complex coefficients. It is known that \operatorname{rd}_{\mathbb{C}}(n)=1 when n\leqslant 5 . Hilbert was particularly interested in the next three cases: he asked if \operatorname{rd}_{\mathbb{C}}(6)=2 (Hilbert’s Sextic conjecture), \operatorname{rd}_{\mathbb{C}}(7)=3 (Hilbert’s 13th problem) and \operatorname{rd}_{\mathbb{C}}(8)=4 (Hilbert’s Octic conjecture). These problems remain open. It is known that \operatorname{rd}_{\mathbb{C}}(6)\leqslant 2 , \operatorname{rd}_{\mathbb{C}}(7)\leqslant 3 and \operatorname{rd}_{\mathbb{C}}(8)\leqslant 4 . It is not known whether or not \operatorname{rd}_{\mathbb{C}}(n) can be >1 for any n\geqslant 6 .In this paper, we show that all three of Hilbert’s conjectures can fail if we replace \mathbb{C} with a base field of positive characteristic.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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) | 0.001 | 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 it