Abstract
We show that for any natural number \(m\), there is an integer \(g_0\ge m\) such that for any \(g\ge g_0\) there exists a function field \(F/\mathbb {F}_q\) of genus \(g\) whose \(L\)-polynomial satisfies
where \(F_1,\ldots , F_{m}\) are the first \(m\) Fibonacci numbers. This fact shows that the set \(\{F_1,\ldots ,F_m\}\) is a non-trivial example of a set of integers satisfying the conditions of a recent theorem by Anbar and Stichtenoth [(Bull Braz Math Soc 44:173–193, 2013), Thm. 8.1] about the first coefficients of the \(L\)-polynomial of a function field.
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References
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We thank the referees for useful comments.
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The second author was partially supported by Project PAPIIT IN104512.
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León-Cardenal, E., Luca, F. \(L\)-polynomials of function fields and Fibonacci Numbers. Bol. Soc. Mat. Mex. 21, 163–169 (2015). https://doi.org/10.1007/s40590-014-0047-1
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DOI: https://doi.org/10.1007/s40590-014-0047-1