Some (non-)elimination results for curves in geometric structures

Serge Randriambololona ¹   ; Sergei Starchenko ²

¹ Department of Mathematics The University of Western Ontario London, Ontario N6A 5B7, Canada
² Department of Mathematics University of Notre Dame Notre Dame, IN 46556, U.S.A.

Fundamenta Mathematicae, Tome 214 (2011) no. 2, pp. 181-198

Résumé

We show that the first order structure whose underlying universe is $\mathbb C$ and whose basic relations are all algebraic subsets of $\mathbb C^2$ does not have quantifier elimination. Since an algebraic subset of $\mathbb C ^2$ is either of dimension $\leq 1$ or has a complement of dimension $\leq 1$, 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 $\mathbb C$ and a predicate for each algebraic subset of $\mathbb C^n$ of dimension $\leq 1$ has quantifier elimination.

Détail

DOI : 10.4064/fm214-2-5

Keywords: first order structure whose underlying universe mathbb whose basic relations algebraic subsets mathbb does have quantifier elimination since algebraic subset mathbb either dimension leq has complement dimension leq restate former result failure quantifier elimination planar complex algebraic curves prove removing planarity hypothesis suffices recover quantifier elimination structure universe mathbb predicate each algebraic subset mathbb dimension leq has quantifier elimination