Geodesic
The complexity of the decision problem for the first order theory of algebraically closed fields
Izvestiya. Mathematics , Tome 29 (1987) no. 2, pp. 459-475

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An algorithm is described that constructs, from every formula of the first order theory of algebraically closed fields, an equivalent quantifier-free formula in time which is polynomial in $\mathscr L^{n^{2a+1}}$, where $\mathscr L$ is the size of the formula, $n$ is the number of variables, and $a$ is the number of changes of quantifiers. Bibliography: 15 titles. 
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     author = {D. Yu. Grigor'ev},
     title = {The complexity of the decision problem for the first order theory of algebraically closed fields},
     journal = {Izvestiya. Mathematics },
     pages = {459--475},
     publisher = {mathdoc},
     volume = {29},
     number = {2},
     year = {1987},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1987_29_2_a9/}
}
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D. Yu. Grigor'ev. The complexity of the decision problem for the first order theory of algebraically closed fields. Izvestiya. Mathematics , Tome 29 (1987) no. 2, pp. 459-475. http://geodesic.mathdoc.fr/item/IM2_1987_29_2_a9/