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Texas A&M Number Theory Seminar

Department of Mathematics
Milner 216
Wednesdays, 1:45-2:45 PM

Martin Widmer

Universität Basel

Wednesday, April 23

Milner 216, 1:45PM

Title:Counting algebraic points of fixed degree and bounded height

Abstract: Given a number field k and natural numbers n and d. The distribution of points in projective space of dimension n over the algebraic closure of k is best described in terms of their height. A well-known result due to Northcott states that the subset of points of degree d (over k) and height not exceeding X is finite for each real number X. A central problem consists in finding an asymptotic estimate for the cardinality of this set as X tends to infinity. A classical theorem of Schanuel gives the asymptotics for d=1. Schmidt (1995), Gao (1996) and more recently Masser and Vaaler (2007) found asymptotic estimates for d>1. Masser and Vaaler's result then covers all cases with n=1; but if k is not the field of rational numbers and n and d are both greater than one not a single example for the asymptotics was known up to now. We present a result which covers the cases n>5d/2+4 for arbitrary number fields k.