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

Serge Randriambololona, and Sergei Starchenko. "Some (non-)elimination results for curves in geometric structures." Fundamenta Mathematicae 214.2 (2011): 181-198. <http://eudml.org/doc/282722>.

@article{SergeRandriambololona2011,
abstract = {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.},
author = {Serge Randriambololona, Sergei Starchenko},
journal = {Fundamenta Mathematicae},
keywords = {quantifier elimination; definable set; geometric structure; constructible curve; definable closure; planar complex algebraic curves},
language = {eng},
number = {2},
pages = {181-198},
title = {Some (non-)elimination results for curves in geometric structures},
url = {http://eudml.org/doc/282722},
volume = {214},
year = {2011},
}

TY - JOUR
AU - Serge Randriambololona
AU - Sergei Starchenko
TI - Some (non-)elimination results for curves in geometric structures
JO - Fundamenta Mathematicae
PY - 2011
VL - 214
IS - 2
SP - 181
EP - 198
AB - 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.
LA - eng
KW - quantifier elimination; definable set; geometric structure; constructible curve; definable closure; planar complex algebraic curves
UR - http://eudml.org/doc/282722
ER -