Zero-free neighborhoods around the unit circle for Kac polynomials
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Bibliographic record
Abstract
Abstract In this paper, we study how the roots of the Kac polynomials $$W_n(z) = \sum _{k=0}^{n-1} \xi _k z^k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>W</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mi>ξ</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mi>z</mml:mi><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:math> concentrate around the unit circle when the coefficients of $$W_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>W</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> are independent and identically distributed nondegenerate real random variables. It is well known that the roots of a Kac polynomial concentrate around the unit circle as $$n\rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math> if and only if $${\mathbb {E}}[\log ( 1+ |\xi _0|)]<\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>E</mml:mi><mml:mo>[</mml:mo><mml:mo>log</mml:mo><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mo>|</mml:mo></mml:mrow><mml:msub><mml:mi>ξ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>|</mml:mo><mml:mo>)</mml:mo><mml:mo>]</mml:mo><mml:mo><</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:mrow></mml:math> . Under the condition $${\mathbb {E}}[\xi ^2_0]<\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>E</mml:mi><mml:mo>[</mml:mo><mml:msubsup><mml:mi>ξ</mml:mi><mml:mn>0</mml:mn><mml:mn>2</mml:mn></mml:msubsup><mml:mo>]</mml:mo><mml:mo><</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math> , we show that there exists an annulus of width $${\text {O}}(n^{-2}(\log n)^{-3})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>O</mml:mtext><mml:mo>(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo>log</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math> around the unit circle which is free of roots with probability $$1-{\text {O}}({(\log n)^{-{1}/{2}}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mtext>O</mml:mtext><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo>log</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math> . The proof relies on small ball probability inequalities and the least common denominator used in [17].
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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 | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.002 | 0.000 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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