On Liouvillian Solutions of Third Order Homogeneous Linear Differential Equations
Bibliographic record
Abstract
In this article we will consider third order homogeneous differential equations: L(y)=y'''+a_1y'+a_0y(a_0,a_1 ∈k) whose Galois group G（L） is imprimitive. This case is characterised by the fact that the third symmetric power equation L ^ⓢ3(y)=0 has an exponential solution whose square is rational (Singer & Ulmer 1993). If L(y)=0 has a Liouvillian solution z whose logarithmic derivative u=z'/z is algebraic over a differential field (k,') ,we will give an algorithm to find the relation between a_0, a_1 , the semi-invariant S=Y_1Y_2Y_3 which is unique up to multiplication by a constant, the coefficients C_0, C_1 of the minimal polynomial P(u) of u and their derivatives. The aim of this work is to diminutize the number of constants C_m stated in the algorithm of Singer & Ulmer (Singer & Ulmer 1993 Algorithm p. 31) whose determination is not easy to do, and we will achieve this by using Groebner Basis.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 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".