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Preprint Number 1408

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1408. Luck Darnière (LAREMA)
Scaled lattices of closed p-adic semi-algebraic sets

Submission date: 3 April 2018


Let p be prime number, K be a p-adically closed field, X⊆ K m a semi-algebraic set defined over K and L(X) the lattice of semi-algebraic subsets of X which are closed in X. We prove that the complete theory of L(X) eliminates the quantifiers in a certain language LASC, the LASC-structure on L(X) being an extension by definition of the lattice structure. Moreover it is decidable, contrary to what happens over a real closed field. We classify these LASC-structures up to elementary equivalence, and get in particular that the complete theory of L(K m) only depends on m, not on K nor even on p. As an application we obtain a classification of semi-algebraic sets over countable p-adically closed fields up to so-called “pre-algebraic” homeomorphisms.

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Full text arXiv 1804.01421: pdf, ps.

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