## 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) — (*n*_{1} + *n*_{2} + 2*N*_{3}) under certain conditions, where *n*_{i} is the number of distinct roots of *f* with multiplicity *i* and *N*_{3} is the number of distinct roots of *f* with multiplicity at least 3.

## Kewywords

complex polynomial of one variable dimension vector space multipliers## Preview

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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

- 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)].CrossRefGoogle 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)].CrossRefGoogle 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).CrossRefGoogle 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).MathSciNetCrossRefGoogle Scholar - 5.E. Rees., “On a paper by Yuri G. Zarhin.,” Eur. J. Math.
**1**(4), 717–720 (2015).MathSciNetCrossRefGoogle Scholar - 6.D. S. Dummit and R. M. Foote,
*Abstract Algebra*(John Wiley & Sons, Hoboken, NJ, 2004).zbMATHGoogle Scholar - 7.V.V. Prasolov,
*Polynomials*, in*Algorithms Comput. Math.*(Springer-Verlag, Berlin, 2010), Vol. 11.Google Scholar