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We solve an energy minimization problem for local fields. As an applicationof these results, we improve on lower bounds set by Bombieri and Zannier forthe limit infimum of the Weil height in fields of totally p-adic numbers andgeneralizations thereof. In the case of fields with mixed archimedean andnon-archimedean splitting conditions, we are able to combine our bounds withsimilar bounds at the archimedean places for totally real fields.

Energy integrals over local fields and global height bounds