Abstract
An exponential polynomial of order q is an entire function of the form
where the coefficients Pj(z),Qj(z) are polynomials in z such that
It is known that the majority of the zeros of a given exponential polynomial are in domains surrounding finitely many critical rays. The shape of these domains is refined by showing that in many cases the domains can approach the critical rays asymptotically. Further, it is known that the zeros of an exponential polynomial are always of bounded multiplicity. A new sufficient condition for the majority of zeros to be simple is found. Finally, a division result for a quotient of two exponential polynomials is proved, generalizing a 1929 result by Ritt in the case q = 1 with constant coefficients. Ritt’s result is closely related to Shapiro’s conjecture that has remained open since 1958.
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Supported by the Academy of Finland #268009, and the Vilho, Yrjö and Kalle Väisälä Fund of the Finnish Academy of Science and Letters.
Supported by the discretionary budget (2016) of the President of the Open University of Japan.
Supported by the JSPS KAKENHI Grant Numbers 16K05194 and 25400131.
Supported by National Natural Science Foundation for the Youth of China (No. 11501402) and Shanxi Scholarship council of China (No. 2015-043).
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Heittokangas, J., Ishizaki, K., Tohge, K. et al. Zero distribution and division results for exponential polynomials. Isr. J. Math. 227, 397–421 (2018). https://doi.org/10.1007/s11856-018-1738-5
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DOI: https://doi.org/10.1007/s11856-018-1738-5