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Proceedings of the London Mathematical Society Advance Access first published online on January 4, 2008
This version published online on June 13, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdm052
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© 2007 London Mathematical Society

Strongly minimal expansions of (C, +) definable in o-minimal fields

Assaf Hasson

Mathematical Institute
Oxford University
24-29 St Giles’
Oxford OX1 3LB
United Kingdom

http://www.maths.ox.ac.uk/~hasson/

Piotr Kowalski

Instytut Matematyczny
Uniwersytet Wroclawski
pl. Grunwaldzki 2/4
50-384 Wroclaw
Poland
pkowa@math.uni.wroc.pl
http://www.math.uni.wroc.pl/~pkowa/

Received 5 December 2006. Accepted 6 September 2007.

We characterize those functions f:C->C definable in o-minimal expansions of the reals for which the structure (C,+, f) is strongly minimal: such functions must be complex constructible, possibly after conjugating by a real matrix. In particular we prove a special case of the Zilber Dichotomy: an algebraically closed field is definable in certain strongly minimal structures which are definable in an o-minimal field.

The first author was supported by the EPSRC grant no. EP C52800X 1, and the second author was supported by a MODNET (European Commission Research Training Network) grant and by the Polish grants: KBN no. 2P03A 018 24 and MEN no. N201 032 32/2231

2000 Mathematics Subject Classification 03C64, 03C45, 14P25.


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