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Auxiliary Polynomials in Number Theory by David Masser

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15

Equidistribution – Bilu

For an algebraic number α of degree d let α1, . . . , αd be the conjugates as in (14.2). After seeing some equidistribution results for conjugates on abelian varieties, Bilu proved the analogue for α (and even for a higher-dimensional α). Now the equidistribution is only relative to the unit circle; roughly speaking, as the height H(α) ≥ 1 gets closer to 1 then the conjugates cluster nearer the unit circle and if in addition d gets large then they are more evenly equidistributed around it.

This is very well illustrated by the example (14.7) with H(21/d) = 21/d as d → ∞, when the conjugates are the image for ζd = e2πi/d. It ...

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