The complexity of the decision problem for the first order theory of algebraically closed fields

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.
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