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Transactions of the American Mathematical Society

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Internality of logarithmic-differential pullbacks


Authors: Ruizhang Jin and Rahim Moosa
Journal: Trans. Amer. Math. Soc. 373 (2020), 4863-4887
MSC (2010): Primary 03C98, 12H05
DOI: https://doi.org/10.1090/tran/8048
Published electronically: April 16, 2020
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Abstract: A criterion in the spirit of Rosenlicht is given, on the rational function $ f(x)$, for when the planar vector field $ \left \{\begin {smallmatrix}y'=xy\\ x'=f(x) \end{smallmatrix}\right \}$ admits a pair of algebraically independent first integrals over some extension of the base field. This proceeds from model-theoretic considerations by working in the theory of differentially closed fields of characteristic zero and asking: If $ D\subseteq \mathbb{A}^1$ is a strongly minimal set internal to the constants, when is $ \operatorname {log}_{\delta }^{-1}(D)$, the pullback of $ D$ under the logarithmic derivative, itself internal to the constants?


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Additional Information

Ruizhang Jin
Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada
Email: r6jin@uwaterloo.ca

Rahim Moosa
Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada
Email: rmoosa@uwaterloo.ca

DOI: https://doi.org/10.1090/tran/8048
Keywords: Logarithmic derivative, internality to the constants, differentially closed fields
Received by editor(s): June 20, 2019
Received by editor(s) in revised form: August 28, 2019, and October 23, 2019
Published electronically: April 16, 2020
Additional Notes: The second author was partially supported by NSERC Discovery and DAS grants.
Article copyright: © Copyright 2020 American Mathematical Society