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Friday, July 16, 2010 |
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Maximum Stirling Numbers of the Second Kind |
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Say an integer n is exceptional if the maximum Stirling number of the
second kind S(n,k) occurs for two (of necessity consecutive) values of k. We prove
that the number of exceptional integers less than or equal to x is
O(xε), for any ε > 0. A theorem of Bombieri and Pila
estimating lattice points on a smooth curve is essential. An estimate for the
number of intersections of an analytic curve with a polynomial of degree d is
also essential. The theory of fewnomials as developed by Khovanskii seems
essential. The exact result we need is found in a paper of Gwozdziewicz,
K. Kurdyka and A. Parusinski estimating the number of solutions of an
algebraic equation on the curve y = ex + sin x, x > 0. J. Pila provided
this reference. |