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A Characterization of Multiplicity-Preserving Global Bifurcations of Complex Polynomial Vector Fields

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Abstract

For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree d, it is proved that any bifurcation which preserves the multiplicity of equilibrium points admits a decomposition into a finite number of elementary bifurcations, and the elementary bifurcations are characterized.

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Notes

  1. For \(d=2\), there are technically no separatrices since infinity is not a pole. The results in this paper still hold in this case if we consider the trajectories through infinity as separatrices.

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Acknowledgements

There are many people to thank for patiently listening to various stages of this work and providing helpful suggestions. In particular, the author is indebted to Bodil Branner, Frederick Gardiner and the other participants in the Extremal Length seminar at CUNY Graduate Center, Christian Henriksen, Poul G. Hjorth, Louis Pedersen, and Tan Lei. The author would also like to express gratitude to the anonymous referee, whose thorough and thoughtful critiques have greatly improved this manuscript.

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  1. Department of Mathematics and Computer Science, Bronx Community College of the City University of New York, 2155 University Avenue, Bronx, NY, 10453, USA

    Kealey Dias

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  1. Kealey Dias
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Correspondence to Kealey Dias.

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This research was supported by Fondation Idella, the Marie Curie European Union Research Training Network Conformal Structures and Dynamics (CODY), the Research Foundation of CUNY PSC-CUNY Cycle 44 (66148-00 44) and Cycle 47 (69510-00 47) Research Awards, the Bronx Community College Foundation Faculty Scholarship Grant 2016, and the Association for Women in Mathematics Travel Grant October 2019 Cycle (NSF 1642548).

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Dias, K. A Characterization of Multiplicity-Preserving Global Bifurcations of Complex Polynomial Vector Fields. Qual. Theory Dyn. Syst. 19, 90 (2020). https://doi.org/10.1007/s12346-020-00424-y

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