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 ℂ and a predicate for each algebraic subset of ℂⁿ of dimension ≤ 1 has quantifier elimination.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-2-5,
author = {Serge Randriambololona and Sergei Starchenko},
title = {Some (non-)elimination results for curves in geometric structures},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {181-198},
zbl = {1252.03081},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-2-5}
}
Serge Randriambololona; Sergei Starchenko. Some (non-)elimination results for curves in geometric structures. Fundamenta Mathematicae, Tome 215 (2011) pp. 181-198. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-2-5/