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Transcendental holomorphic maps between real algebraic manifolds in a complex space


Author: Guillaume Rond
Journal: Proc. Amer. Math. Soc. 148 (2020), 2097-2102
MSC (2010): Primary 32H02; Secondary 05A15, 14P05, 32C05, 32V40, 39B32
DOI: https://doi.org/10.1090/proc/14865
Published electronically: January 15, 2020
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Abstract: We give an example of a real algebraic manifold embedded in a complex space that does not satisfy the Nash-Artin approximation property. This Nash-Artin approximation property is closely related to the problem of determining when the biholomorphic equivalence for germs of real algebraic manifolds coincides with the algebraic equivalence. This example is an elliptic Bishop surface, and its construction is based on the functional equation satisfied by the generating series of some walks restricted to the quarter plane.


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Additional Information

Guillaume Rond
Affiliation: Aix-Marseille Université, LASOL, UMI2001, UNAM, Mexico
Email: guillaume.rond@univ-amu.fr

DOI: https://doi.org/10.1090/proc/14865
Keywords: Holomorphic map, algebraic map, Bishop surface, lattice walk
Received by editor(s): May 21, 2019
Received by editor(s) in revised form: September 2, 2019, and September 17, 2019
Published electronically: January 15, 2020
Additional Notes: The author is deeply grateful to the UMI LASOL of the CNRS where this project has been carried out.
Communicated by: Harold P. Boas
Article copyright: © Copyright 2020 American Mathematical Society