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CELL DECOMPOSITION AND CLASSIFICATION OF DEFINABLE SETS IN p-OPTIMAL FIELDS

Published online by Cambridge University Press:  24 January 2017

LUCK DARNIÈRE
Affiliation:
FACULTÉ DES SCIENCES 2 BOULEVARD LAVOISIER 49045 ANGERS CEDEX 01, FRANCE
IMMANUEL HALPUCZOK
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF LEEDS WOODHOUS LANE, LEEDS, UK

Abstract

We prove that for p-optimal fields (a very large subclass of p-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strong p-minimality. Then we turn to strongly p-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to a p-adic one. For such fields K, we prove that every definable subset of K × Kd whose fibers over K are inverse images by the valuation of subsets of the value group is semialgebraic. Combining the two we get a preparation theorem for definable functions on p-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension.

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Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

REFERENCES

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