Submission date: 17 May 2011.

Abstract:

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

Mathematics Subject Classification: 03C10, 03C60, 14H50

Keywords and phrases:

Full text arXiv 1011.2432: pdf, ps.