The euclidean minimum of a number field is a real number, which measureshow far the
number field is from being euclidean with respect to the norm. A still open conjecture of
Minkowski predicts an upper bound for the euclidean minima of totally real number
fields in terms of the degree and the discriminant.
I will explain how a topological method due to McMullen can be used to produce upper
bounds for euclidean minima of arbitrary number fields in terms of signature and discriminant.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/246