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

TL;DR

The paper investigates the limits of quantifier elimination in structures based on complex algebraic sets, showing failure in the planar case and success when generalizing to all dimensions.

Contribution

It demonstrates that quantifier elimination fails for structures with algebraic relations in the plane but holds when extended to all algebraic subsets of dimension at most one.

Findings

Quantifier elimination fails for the structure with algebraic subsets of $ ext{C}^2$.

Quantifier elimination holds for the structure with algebraic subsets of $ ext{C}^n$ of dimension ≤ 1.

The result clarifies the role of planarity in algebraic structures and their logical properties.

Abstract

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