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Spaces of Polynomials Related to Multiplier Maps

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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.

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References

  1. Yu. G. Zarhin, “Polynomials in one variable and ranks of certain tangent maps,” Mat. Zametki 1 (4), 539–550 (2012) [Math. Notes 1(4), 508–516 (2012)].

    Article  Google Scholar 

  2. Yu. G. Zarhin, “One-dimensional polynomial maps, periodic points and multipliers,” Izv. Ross. Akad. Nauk Ser. Mat. 1 (4), 59–72 (2013) [Izv. Math. 1 (4), 700–713 (2013)].

    Article  Google Scholar 

  3. G. T. Buzzard, S. L. Hruska, and Yu. Ilyashenko., “Kupka-Smale theorem for polynomial automorphisms of C2 and persistence of heteroclinic intersections,” Invent. Math. 1 (1), 45–89 (2005).

    Article  Google Scholar 

  4. I. Gorbovickis., “Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable,” Ergodic Theory Dynam. Systems 36 (4), 1156–1166(2016).

    Article  MathSciNet  Google Scholar 

  5. E. Rees., “On a paper by Yuri G. Zarhin.,” Eur. J. Math. 1 (4), 717–720 (2015).

    Article  MathSciNet  Google Scholar 

  6. D. S. Dummit and R. M. Foote, Abstract Algebra (John Wiley & Sons, Hoboken, NJ, 2004).

    MATH  Google Scholar 

  7. V.V. Prasolov, Polynomials, in Algorithms Comput. Math. (Springer-Verlag, Berlin, 2010), Vol. 11.

    Google Scholar 

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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.

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Authors and Affiliations

  1. John Hopkins University, Baltimore, 21218, USA

    Zhaoning Yang

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  1. Zhaoning Yang
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Correspondence to Zhaoning Yang.

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The text was submitted by the author in English.

Published in Russian in Matematicheskie Zametki, 2019, Vol. 106, No. 3, pp. 350–376.

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Yang, Z. Spaces of Polynomials Related to Multiplier Maps. Math Notes 106, 342–363 (2019). https://doi.org/10.1134/S0001434619090049

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  • DOI: https://doi.org/10.1134/S0001434619090049

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