Transasymptotic expansions of o-minimal germs
arxiv(2024)
摘要
Given an o-minimal expansion ℝ_𝒜 of the real ordered
field, generated by a generalized quasianalytic class 𝒜, we
construct an explicit truncation closed ordered differential field embedding of
the Hardy field of the expansion ℝ_𝒜,exp of
ℝ_𝒜 by the unrestricted exponential function, into the
field 𝕋 of transseries. We use this to prove some non-definability
results. In particular, we show that the restriction to the positive half-line
of Euler's Gamma function is not definable in the structure
ℝ_an^*,exp, generated by all convergent generalized
power series and the exponential function, thus establishing the
non-interdefinability of the restrictions to a neighbourhood of +∞ of
Euler's Gamma and of the Riemann Zeta function.
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