Geodesic
On elementary theories of algebraically closed groups
Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 186-199

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In paper for any algebraically closed group $G$, as well as for the class of the algebraically closed groups, we prove algorithmic undecidability of the positive $\forall^2 \exists^{24}$-theory and $\forall^3 \exists^{2}$-theory. For an arbitrary $g\in G$, we also prove the decidability of the equation of the type $$ w(x_1, \ldots , x_n) = g, $$ where $w(x_1, \ldots , x_n)$ is a non-empty irreducible word in the unknowns $x_1,\ldots x_n\in G$.
Keywords: algebraically closed group, positive theory, equation. 
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     author = {V. G. Durnev and O. V. Zetkina and A. I. Zetkina},
     title = {On elementary theories of algebraically closed groups},
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V. G. Durnev; O. V. Zetkina; A. I. Zetkina. On elementary theories of algebraically closed groups. Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 186-199. http://geodesic.mathdoc.fr/item/CHEB_2020_21_1_a10/