Shapes, fingerprints and rational lemniscates
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- by Malik Younsi
- Proc. Amer. Math. Soc. 144 (2016), 1087-1093
- DOI: https://doi.org/10.1090/proc12751
- Published electronically: June 30, 2015
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Abstract:
It has been known for a long time that any smooth Jordan curve in the plane can be represented by its so-called fingerprint, an orientation preserving smooth diffeomorphism of the unit circle onto itself. In this paper, we give a new, simple proof of a theorem of Ebenfelt, Khavinson and Shapiro stating that the fingerprint of a polynomial lemniscate of degree $n$ is given by the $n$-th root of a Blaschke product of degree $n$ and that, conversely, any smooth diffeomorphism induced by such a map is the fingerprint of a polynomial lemniscate of the same degree. The proof is easily generalized to the case of rational lemniscates, thus solving a problem raised by the previously mentioned authors.References
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Bibliographic Information
- Malik Younsi
- Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
- MR Author ID: 1036614
- Email: malik.younsi@gmail.com
- Received by editor(s): June 13, 2014
- Received by editor(s) in revised form: February 6, 2015
- Published electronically: June 30, 2015
- Additional Notes: This research was supported by NSERC
- Communicated by: Jeremy Tyson
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1087-1093
- MSC (2010): Primary 37E10, 30C20; Secondary 30F10
- DOI: https://doi.org/10.1090/proc12751
- MathSciNet review: 3447662