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Level curves of rational functions and unimodular points on rational curves


Authors: Fedor Pakovich and Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 148 (2020), 1829-1833
MSC (2010): Primary 11D61, 12D10, 30C15, 30J10
DOI: https://doi.org/10.1090/proc/14928
Published electronically: January 21, 2020
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Abstract: We obtain an improvement and broad generalisation of a result of N. Ailon and Z. Rudnick (2004) on common zeros of shifted powers of polynomials. Our approach is based on reducing this question to a more general question of counting intersections of level curves of complex functions. We treat this question via classical tools of complex analysis and algebraic geometry.


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Additional Information

Fedor Pakovich
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Beer Sheva, 8410501, Israel
Email: pakovich@math.bgu.ac.il

Igor E. Shparlinski
Affiliation: Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
Email: igor.shparlinski@unsw.edu.au

DOI: https://doi.org/10.1090/proc/14928
Keywords: Unimodular points, Ailon and Rudnick theorem, Blaschke product
Received by editor(s): May 16, 2018
Received by editor(s) in revised form: July 29, 2018
Published electronically: January 21, 2020
Additional Notes: The work of the second author was supported in part by the Australian Research Council Grants DP170100786 and DP180100201.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2020 American Mathematical Society