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Auxiliary Polynomials in Number Theory by David Masser

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17

Height upper bounds

After the relatively easy lower bound Theorem 14.8 and the really easy upper bound Theorem 14.9, together with the distinctly tricky lower bound Theorem 16.1, it seems only fair on grounds of symmetry to present a more difficult upper bound. The Mirimanov polynomials are defined as Mn(Y) = (Y + 1)nYn − 1 deprived of some obvious factors, and in the course of an investigation into their irreducibility properties, Beukers proved that if n ≥ 2 then the absolute height H(β) ≤ 216 for any zero β. He used hypergeometric functions. To make this look more like our previous results, we formulate it for

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