### On the Elementary Theory of Restricted Real and Imaginary Parts of Holomorphic Functions

Hassan Sfouli
Source: Notre Dame J. Formal Logic Volume 53, Number 1 (2012), 67-77.

#### 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.

First Page:
Primary Subjects: 03C10
Secondary Subjects: 14P15
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Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1336586238
Digital Object Identifier: doi:10.1215/00294527-1626527
Zentralblatt MATH identifier: 06040395
Mathematical Reviews number (MathSciNet): MR2925269

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