Energy integrals over local fields and global height bounds

Paul Fili, Clayton Petsche

Published 2013-06-15, updated 2014-10-11Version 3

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