Geodesic
Spaces of Polynomials Related to Multiplier Maps
Matematičeskie zametki, Tome 106 (2019) no. 3, pp. 350-376

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Let $f(x)$ be a complex polynomial of degree $n$. We associate $f$ with a $\mathbb{C}$-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 $(n-1)-(n_1+n_2+2N_3)$ 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 $3$.
Mots-clés : complex polynomial of one variable, dimension, multipliers.
Keywords: vector space 
@article{MZM_2019_106_3_a3,
     author = {Zhaoning Yang},
     title = {Spaces of {Polynomials} {Related} to {Multiplier} {Maps}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {350--376},
     publisher = {mathdoc},
     volume = {106},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_106_3_a3/}
}
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Zhaoning Yang. Spaces of Polynomials Related to Multiplier Maps. Matematičeskie zametki, Tome 106 (2019) no. 3, pp. 350-376. http://geodesic.mathdoc.fr/item/MZM_2019_106_3_a3/