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Bulletin of the London Mathematical Society Advance Access published online on March 20, 2008
Bulletin of the London Mathematical Society, doi:10.1112/blms/bdn004
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© 2008 London Mathematical Society

Locally polynomially bounded structures

G. O. Jones

Mathematical Institute
24–29 St Giles’
Oxford
OX1 3LB
United Kingdom
Current address:
Department of Mathematics and Statistics
McMaster University
1280 Main Street West
Hamilton
Ontario
Canada L8S 4K1

A. J. Wilkie

Mathematical Institute
24–29 St Giles’
Oxford
OX1 3LB
United Kingdom
Current address:
School of Mathematics
The Alan Turing Building
University of Manchester
Manchester
M13 9PL
United Kingdom
awilkie@maths.man.ac.uk

Received 18 January 2007. Revision received 7 November 2007.

We prove a theorem which provides a method for constructing points on varieties defined by certain smooth functions. We require that the functions be definable in a definably complete expansion of a real closed field and be locally definable in a fixed o-minimal and polynomially bounded reduct. As an application we show that in certain o-minimal structures, definable functions are piecewise implicitly defined over the basic functions in the language.

2000 Mathematics Subject Classification 03C64.

The first author was supported by the EPSRC.


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