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| [[File:Polynome de fekete 43.svg|thumbnail|300px|right|Roots of the Fekete polynomial for p = 43]]
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| In [[mathematics]], a '''Fekete polynomial''' is a [[polynomial]]
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| :<math>f_p(t):=\sum_{a=0}^{p-1} \left (\frac{a}{p}\right )t^a\,</math>
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| where <math>\left(\frac{\cdot}{p}\right)\,</math> is the [[Legendre symbol]] modulo some integer ''p'' > 1.
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| These polynomials were known in nineteenth-century studies of [[Dirichlet L-function]]s, and indeed to [[Peter Gustav Lejeune Dirichlet]] himself. They have acquired the name of [[Michael Fekete]], who observed that the absence of real zeroes ''a'' of the Fekete polynomial with 0 < ''a'' < 1 implies an absence of the same kind for the [[L-function]]
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| :<math> L\left(s,\dfrac{x}{p}\right).\, </math>
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| This is of considerable potential interest in [[number theory]], in connection with the hypothetical [[Siegel zero]] near ''s'' = 1. While numerical results for small cases had indicated that there were few such real zeroes, further analysis reveals that this may indeed be a 'small number' effect.
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| ==References==
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| * [[Peter Borwein]]: ''Computational excursions in analysis and number theory.'' Springer, 2002, ISBN 0-387-95444-9, Chap.5.
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| ==External links==
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| * [[Brian Conrey]], [[Andrew Granville]], [[Bjorn Poonen]] and [[Kannan Soundararajan]], ''[http://arxiv.org/abs/math/9906214v1 Zeros of Fekete polynomials]'', [[arXiv]] e-print math.NT/9906214, June 16, 1999.
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| [[Category:Polynomials]]
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| [[Category:Zeta and L-functions]]
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Revision as of 13:32, 11 February 2014
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