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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.