On the Koebe quarter theorem for trinomials with fold symmetry
Bibliographic record
Abstract
The Koebe problem for univalent polynomials with real coefficients is fully solved only for trinomials, which means that in this case the Koebe radius and the extremal polynomial (extremizer) have been found. The general case remains open, but conjectures have been formulated. The corresponding conjectures have also been hypothesized for univalent polynomials with real coefficients and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -fold rotational symmetry. This paper provides confirmation of these hypotheses for trinomials <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z plus a z Superscript upper T plus 1 plus b z Superscript 2 upper T plus 1"> <mml:semantics> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:msup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>T</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:msup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">z + az^{T + 1} + bz^{2T + 1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Namely, the Koebe radius is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r equals 4 cosine squared left-parenthesis StartFraction pi left-parenthesis 1 plus upper T right-parenthesis Over 2 plus 3 upper T EndFraction right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> <mml:msup> <mml:mi>cos</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> </mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi> π </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> <mml:mi>T</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">r=4\cos ^2\left (\frac {\pi (1+T)}{2+3T}\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and the only extremizer of the Koebe problem is the trinomial <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper B Superscript left-parenthesis upper T right-parenthesis Baseline left-parenthesis z right-parenthesis equals z plus StartFraction 2 Over 2 plus 3 upper T EndFraction left-parenthesis negative upper T plus left-parenthesis 2 plus 2 upper T right-parenthesis cosine left-parenthesis StartFraction pi upper T Over 2 plus 3 upper T EndFraction right-parenthesis right-parenthesis z Superscript 1 plus upper T Baseline plus 2nd Row plus StartFraction 1 Over 2 plus 3 upper T EndFraction left-parenthesis 2 plus upper T minus 2 upper T cosine left-parenthesis StartFraction pi upper T Over 2 plus 3 upper T EndFraction right-parenthesis right-parenthesis z Superscript 1 plus 2 upper T Baseline period EndLayout"> <mml:semantics> <mml:mtable rowspacing="3pt" columnspacing="1em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:msup> <mml:mi>B</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>z</mml:mi> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> <mml:mi>T</mml:mi> </mml:mrow> </mml:mfrac> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo> − </mml:mo> <mml:mi>T</mml:mi> <mml:mo>+</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>cos</mml:mi> <mml:mo> </mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi> π </mml:mi> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> <mml:mi>T</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| 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".