Abstract
We show that the ordered field of real numbers with restricted $\mathbb{R}_{\mathscr{H}}$-definable analytic functions admits quantifier elimination if we add a function symbol $^{-1}$ for the function $x\mapsto \frac{1}{x}$ (with $0^{-1}=0$ by convention), where $\mathbb{R}_{\mathscr{H}}$ is the real field augmented by the functions in the family $\mathscr{H}$ of restricted parts (real and imaginary) of holomorphic functions which satisfies certain conditions. Further, with another condition on $\mathscr{H}$ we show that the structure ($\mathbb{R}_{\mathscr{H}}$, constants) is strongly model complete.
Citation
Hassan Sfouli. "On the Elementary Theory of Restricted Real and Imaginary Parts of Holomorphic Functions." Notre Dame J. Formal Logic 53 (1) 67 - 77, 2012. https://doi.org/10.1215/00294527-1626527
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