On Families QSL $$_{\ge 2}$$ of Quadratic Systems with Invariant Lines of Total Multiplicity At Least 2
Bibliographic record
Abstract
Let QSL $$_{\ge i}$$ be the family of quadratic differential systems with invariant lines of total multiplicity at least i and let QSL $$_{i}$$ denote the family of quadratic systems with invariant lines of total multiplicity exactly i. For any polynomial system the line at infinity is invariant. Thus the family $${{\varvec{QS}}}\,$$ of all quadratic systems is the same as QSL $$_{\ge 1}$$ . Our main interest is in the family QSL $$_{\ge 2}$$ , the largest strict sub-family of $${{\varvec{QS}}}\,$$ of systems having invariant lines. Our first goal is to give a brief survey of what we know so far about this family. At the moment we know everything about the family QSL $$_{\ge 4}$$ as well as about two important subfamilies of QSL $$_{3}$$ . Only two other subfamilies of QSL $$_{3}$$ remain to be studied. Our second goal is to give necessary and sufficient conditions in terms of invariant polynomials for systems to belong to one of these, namely the family QSL $$^{2p}$$ of quadratic systems with two parallel lines real or complex and to give the geometrical classification of this family. Our third goal is to complete a former partial result in the literature and obtain the global result on QSL $$_{\ge 2}$$ saying that the Hilbert number for this family is one. This opens the road for attempting to obtain the topological classification of QSL $$_{\ge 2}$$ .
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.003 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.016 | 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".