2024/12/26 by Valentin Huguin, Huguin, Valentin · 4 citations
Mathematics · Computer Science · #Advanced Differential Equations and Dynamical Systems #advanced mathematical theories #Coding theory and cryptography
paper · pdf · doi:10.48550/arxiv.2412.19335
Fix an integer d ≥ 2. The space Pd of polynomial maps of degree d modulo conjugation by affine transformations is naturally an affine variety over ℚ of dimension d -1. For each integer P ≥ 1, the elementary symmetric functions of the multipliers at all the cycles with period p ∈ \lbrace 1, \dotsc, P \rbrace induce a natural morphism Multd(P) defined on Pd. In this article, we show that the morphism Multd(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 Multd(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 ℂ of degree d with bounded multipliers at its cycles with periods 1 and 2 is necessarily bounded in Pd(ℂ). \mathord\bullet A generic conjugacy class of polynomials over ℂ of degree d is uniquely determined by its multipliers at its cycles with periods 1 and 2.