2024/02/19 by Rolin, Jean-Philippe, Servi, Tamara, Speissegger, Patrick
#03C10 #03C64 #12J15 #26E10 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2402.12073
Given an o-minimal expansion ℝA 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 ℝA,exp of ℝA by the unrestricted exponential function, into the field \mathbbT 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 ℝan*,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.