Integral Points of Fixed Degree and Bounded Height

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Abstract

By Northcott's Theorem there are only finitely many algebraic points in affine n-space of fixed degree e over a given number field and of height at most X. Finding the asymptotics for these cardinalities as X becomes large is a long-standing problem which is solved only for e=1 by Schanuel, for n=1 by Masser and Vaaler, and for n “large enough” by Schmidt, Gao, and the author. In this paper, we study the case where the coordinates of the points are restricted to algebraic integers, and we derive the analogues of Schanuel's, Schmidt's, Gao's, and the author's results. The proof invokes tools from dynamics on homogeneous spaces, algebraic number theory, geometry of numbers, and a geometric partition method due to Schmidt.
Original languageEnglish
Pages (from-to)3906-3943
Number of pages38
JournalInternational Mathematics Research Notices
Volume2016
Issue number13
Early online date25 Sept 2015
DOIs
Publication statusPublished - 2016

Keywords

  • integral points, height, counting, Pisot numbers

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