Some (non-)elimination results for curves in geometric structures
Tom 214 / 2011
Fundamenta Mathematicae 214 (2011), 181-198 MSC: 03C10, 03C60, 14H50. DOI: 10.4064/fm214-2-5
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.