Spaces of polynomials related to multiplier maps

Abstract

Let $f(x)\in \mathbb{C}[x]$ of degree $n$. We attach to $f$ a $\mathbb{C}$-vector space $W(f)$ which consists of complex polynomials $p(x)$ of degree at most $n-2$ such that $f(x)$ divides $f"(x)p(x)-f^{\prime }(x)p^{\prime }(x)$. The space $W(f)$ originally appears in Yuri Zarhin's solution towards a problem of dynamics in one complex variable posed by Yu. S. Ilyashenko. In this paper, we show that $W(f)$ is nonvanishing if and only if $q(x{)}^2$ divides $f(x)$ for some quadratic polynomial $q(x)$. Then we prove $W(f)$ 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 three.

15 pages

References in corpus (1)

• Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable