Sentences with three quantifiers are decidable in set theory
Separably Real Closed Local Rings.
Some (non-)elimination results for curves in geometric structures
We show that the first order structure whose underlying universe is ℂ and whose basic relations are all algebraic subsets of ℂ² does not have quantifier elimination. Since an algebraic subset of ℂ² is either of dimension ≤ 1 or has a complement of dimension ≤ 1, one can restate the former result as a failure of quantifier elimination for planar complex algebraic curves. We then prove that removing the planarity hypothesis suffices to recover quantifier elimination: the structure with the universe...
Sur la complexité du principe de Tarski-Seidenberg
Sur la complexité du principe de Tarski-Seidenberg