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 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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".