Spaces of Polynomials Related to Multiplier Maps

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 $3$.