An extremal property of Fekete polynomials
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Abstract
The Fekete polynomials are defined as <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript q Baseline left-parenthesis z right-parenthesis colon equals sigma-summation Underscript k equals 1 Overscript q minus 1 Endscripts left-parenthesis StartFraction k Over q EndFraction right-parenthesis z Superscript k"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:=</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:munderover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mi>k</mml:mi> <mml:mi>q</mml:mi> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">F_q(z) := \sum ^{q-1}_{k=1} \left (\frac {k}{q}\right ) z^k</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis StartFraction dot Over q EndFraction right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mo> ⋅ </mml:mo> <mml:mi>q</mml:mi> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left (\frac {\cdot }{q}\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Legendre symbol. These polynomials arise in a number of contexts in analysis and number theory. For example, after cyclic permutation they provide sequences with smallest known <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 4"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">L_4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm out of the polynomials with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="plus-or-minus 1"> <mml:semantics> <mml:mrow> <mml:mo> ± </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\pm 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coefficients. The main purpose of this paper is to prove the following extremal property that characterizes the Fekete polynomials by their size at roots of unity. <bold>Theorem 0.1.</bold> Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals a 1 x plus a 2 x squared plus midline-horizontal-ellipsis plus a Subscript upper N minus 1 Baseline x Superscript upper N minus 1"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">f(x)=a_1x+a_2x^2+\cdots +a_{N-1}x^{N-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with odd <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript n Baseline equals plus-or-minus 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo> ± </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">a_n=\pm 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . If <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max left-brace right-brace colon vertical-bar vertical-bar of ff left-parenthesis righ
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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