Refinements of the Gauss-Lucas theorem using rational lemniscates and polar convexity
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Bibliographic record
Abstract
The classical Gauss-Lucas theorem for complex polynomials <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis z right-parenthesis colon-equal left-parenthesis z minus z 1 right-parenthesis midline-horizontal-ellipsis left-parenthesis z minus z Subscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≔</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p(z) ≔(z-z_1) \cdots (z-z_n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> states that the critical points of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are in the convex hull of its zeros. We give two refinements of the Gauss-Lucas theorem, both connected by the notion of polar convexity. In the first, we describe lemniscatic regions that do not contain non-trivial critical points of any polynomial of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis z minus z 1 right-parenthesis midline-horizontal-ellipsis left-parenthesis z minus z Subscript m Baseline right-parenthesis left-parenthesis z minus z Subscript m plus 1 Superscript asterisk Baseline right-parenthesis midline-horizontal-ellipsis left-parenthesis z minus z Subscript n Superscript asterisk Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo> − </mml:mo> <mml:msubsup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo> − </mml:mo> <mml:msubsup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(z-z_1) \cdots (z-z_m)(z-z^*_{m+1})\cdots (z-z^*_{n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , when the parameters <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z Subscript m plus 1 Superscript asterisk Baseline comma ellipsis comma z Subscript n Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">z^*_{m+1},\ldots ,z^*_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vary freely in a specified set, containing the zeros <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z Subscript m plus 1 Baseline comma ellipsis comma z Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>z</mml:mi>
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| 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 it