Spaces of Polynomials Related to Multiplier Maps
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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.
Kewywordscomplex polynomial of one variable dimension vector space multipliers
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This note was written in an attempt to answer questions suggested by Yuri Zarhin in connection with . 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.
- 7.V.V. Prasolov, Polynomials, in Algorithms Comput. Math. (Springer-Verlag, Berlin, 2010), Vol. 11.Google Scholar