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Auxiliary Polynomials in Number Theory by David Masser

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16

Height lower bounds – Dobrowolski

In 1933 Lehmer, in connexion with the factorization of large integers, looked at integer sequences

image

in the notation of (14.2) with a0 = 1. It was advantageous to have as slow growth as possible. Clearly from (14.3) we have |un| ≤ 2dH(α)dn for the height H(α) and so it would be nice to make H(α)d as small as possible. Of course this can be achieved with roots of unity (or zero) by (14.5), but then un is uninteresting. Lehmer found the value

H(α)d = 1.17628081825991750654407033847403505069341580656469 . . .

(which I calculated by Maple with Newton’s Method (12.11) after three iterations starting with Maple’s first ...

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