Real closed exponential fields

In an extended abstract Ressayre considered real closed exponential fields and integer parts that respect the exponential function. He outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayre's construction, which becomes canonical once we fix the real closed exponential field, a residue field section, and a well ordering of the field. The procedure is constructible over these objects; each step looks effective, but may require many steps. We produce an example of an exponential field $R$ with a residue field $k$ and a well ordering $<$ such that $D^c(R)$ is low and $k$ and $<$ are $\Delta^0_3$, and Ressayre's construction cannot be completed in $L_{\omega_1^{CK}}$.Comment: 24 page

Models of true arithmetic are integer parts of nice real closed fields

Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an exponential real closed field that is elementary equivalent to the reals with exponentiation

Support of Laurent series algebraic over the field of formal power series

This work is devoted to the study of the support of a Laurent series in several variables which is algebraic over the ring of power series over a characteristic zero field. Our first result is the existence of a kind of maximal dual cone of the support of such a Laurent series. As an application of this result we provide a gap theorem for Laurent series which are algebraic over the field of formal power series. We also relate these results to diophantine properties of the fields of Laurent series.Comment: 31 pages. To appear in Proc. London Math. So