Spaces of Polynomials Related to Multiplier Maps

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) — (n₁ + n₂ + 2N₃) under certain conditions, where n_i 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.