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Bibliographic record
Abstract
We consider properties and center conditions for plane polynomial systems of the forms<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mover accent="true"><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi>y</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle="italic">1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle="italic">2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mover accent="true"><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle="italic">1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle="italic">2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle="italic">1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle="italic">1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle="italic">2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle="italic">2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>are polynomials of degrees<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mn fontstyle="italic">2</mml:mn><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn fontstyle="italic">1</mml:mn></mml:math>, respectively, for integers<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn fontstyle="italic">2</mml:mn></mml:math>. We restrict our attention to those systems for which<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mi>y</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle="italic">2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle="italic">2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn fontstyle="italic">0</mml:mn></mml:math>. In this case the system can be transformed to a trigonometric Abel equation which is similar in form to the one obtained for homogeneous systems<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle="italic">2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn fontstyle="italic">2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn fontstyle="italic">0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>. From this we show that any center condition of a homogeneous system for a given<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math>can be transformed to a center condition of the corresponding generalized cubic system and we use a similar idea to obtain center conditions for several other related systems. As in the case of the homogeneous system, these systems can also be transformed to Abel equations having rational coefficients and we briefly discuss an application of this to a particular Abel equation.
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 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