Uniform model-completeness for the real field expanded by power functions

Journal of Symbolic Logic 75 (4):1441-1461 (2010)
Abstract
We prove that given any first order formula φ in the language L' = {+,., <, (f i ) i ∈ I , (c i ) i ∈ I }, where the f i are unary function symbols and the c i are constants, one can find an existential formula ψ such that φ and ψ are equivalent in any L'-structure $\langle {\Bbb R},+,.,<,(x^{c_{i}})_{i\in I},(c_{i})_{i\in I}\rangle $
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