A Nullstellensatz for Henselian fields with real-closed residue field
by
Rafel Farré
Universitat Politècnica de Catalunya

We provide a Nullstellensatz which applies to any Henselian field with real-closed residue field. This result extends all (known to us) previous results for such kind of fields, in particular the well-known results of Krivine-Dubois-Rissler-Stengle (real-closed fields), Jacob [J] (R((t)) and real-series closed fields) and Becker-Jacob [B-J] (generalized real-closed fields).

The main tools are three. First, the Ax-Kochen-Ershov Theorem, which allows us to reduce the problem of existentially closed embeddings from fields to ordered abelian groups. Second, the algebraic characeterization of existentielly closed embeddings of ordered abelian groups obtained in [D-F]. Third a collection of polynomials which encode in the pure field structure the valuation theoretic characterization provided in [D-F]. The work is devoted to the developement of the third part.

References

[B-J] E. Becker and B. Jacob, Rational points on algebraic varieties over a generalized real closed field: a model-theoretic approach, CRELLE 357 (1985) pp.77-95.

[J] B. Jacob, A Nullstellensatz for R((t)), Comunications in Algebra 8 (1980), pp. 1083-1094.

[D-F] F. Delon and R. Farré, existentially closed embeddings of ordered abelian groups, preprint, 1999.

Date received: June 7, 1999