Notre Dame Journal of Formal Logic

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

Hassan Sfouli

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.

Article information

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

Dates
First available in Project Euclid: 9 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1336586238

Digital Object Identifier
doi:10.1215/00294527-1626527

Mathematical Reviews number (MathSciNet)
MR2925269

Zentralblatt MATH identifier
1258.03034

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

Keywords
quantifier elimination Weierstrass systems

Citation

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. https://projecteuclid.org/euclid.ndjfl/1336586238


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References

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