Notre Dame Journal of Formal Logic

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

Hassan Sfouli


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.

Article information

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

First available in Project Euclid: 9 May 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C10: Quantifier elimination, model completeness and related topics
Secondary: 14P15: Real analytic and semianalytic sets [See also 32B20, 32C05]

quantifier elimination Weierstrass systems


Sfouli, Hassan. On the Elementary Theory of Restricted Real and Imaginary Parts of Holomorphic Functions. Notre Dame J. Formal Logic 53 (2012), no. 1, 67--77. doi:10.1215/00294527-1626527.

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