Moduli spaces of polynomial maps and multipliers at small cycles

Fix an integer $d \geq 2$. The space $\mathcal{P}_{d}$ of polynomial maps of degree $d$ modulo conjugation by affine transformations is naturally an affine variety over $\mathbb{Q}$ of dimension $d -1$. For each integer $P \geq 1$, the elementary symmetric functions of the multipliers at all the cycles with period $p \in \lbrace 1, \dotsc, P \rbrace$ induce a natural morphism $\operatorname{Mult}_{d}^{(P)}$ defined on $\mathcal{P}_{d}$. In this article, we show that the morphism $\operatorname{Mult}_{d}^{(2)}$ induced by the multipliers at the cycles with periods $1$ and $2$ is both finite and birational onto its image. In the case of polynomial maps, this strengthens results by McMullen and by Ji and Xie stating that $\operatorname{Mult}_{d}^{(P)}$ is quasifinite and birational onto its image for all sufficiently large integers $P$. Our result arises as the combination of the following two statements: $\mathord{\bullet}$ A sequence of polynomials over $\mathbb{C}$ of degree $d$ with bounded multipliers at its cycles with periods $1$ and $2$ is necessarily bounded in $\mathcal{P}_{d}(\mathbb{C})$. $\mathord{\bullet}$ A generic conjugacy class of polynomials over $\mathbb{C}$ of degree $d$ is uniquely determined by its multipliers at its cycles with periods $1$ and $2$.