Definability of geometric properties in algebraically closed fields

Abstract : We prove that there exists no sentence F of the language of rings with an extra binary predicat I_2 satisfying the following property: for every definable set X of C^2, X is connected if and only if (C,X) satisfies F, where I_2 is interpreted by X. We conjecture that the same result holds for the closed subsets of C^2. We prove some results motivated by this conjecture.
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Pascal Koiran, Olivier Chapuis. Definability of geometric properties in algebraically closed fields. [Research Report] LIP RR-1998-32, Laboratoire de l'informatique du parallélisme. 1998, 2+25p. ⟨hal-02102101⟩

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