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Mathematical Notes

, Volume 106, Issue 3–4, pp 342–363 | Cite as

Spaces of Polynomials Related to Multiplier Maps

  • Zhaoning YangEmail author
Article
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Abstract

Let f(x) be a complex polynomial of degree n. We associate f with a ℂ-vector space W(f) that consists of complex polynomials p(x) of degree at most n — 2 such that f(x) divides f”(x)p(x) — f’(x)p’(x). The space W(f) first appeared in Yu. G. Zarhin’s work, where a problem concerning dynamics in one complex variable posed by Yu. S. Ilyashenko was solved. In this paper, we show that W(f) is nonvanishing if and only if q(x)2 divides f(x) for some quadratic polynomial q(x). In that case, W(f) has dimension (n — 1) — (n1 + n2 + 2N3) under certain conditions, where ni is the number of distinct roots of f with multiplicity i and N3 is the number of distinct roots of f with multiplicity at least 3.

Kewywords

complex polynomial of one variable dimension vector space multipliers 

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Notes

Acknowledgments

This note was written in an attempt to answer questions suggested by Yuri Zarhin in connection with [1]. I would like to thank Yu. G. Zarhin for his questions, stimulating discussions, and interest in this paper. I am also grateful to his patience in reading several preliminary versions of this note and making extremely useful remarks. In addition, I would like to thank Xiyuan Wang, whose comments have helped to improve the exposition.

References

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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.John Hopkins UniversityBaltimoreUSA

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