Irreducibility of eventually 2-periodic curves in the moduli space of cubic polynomials

2023/05/31 by Niladri Patra, Patra, Niladri · 1 citation
Computer Science · Mathematics · #11R09 (Primary) #37F12 #37P45 (Secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2305.19944

openalex publication_date 2023/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the moduli space, M₃, of cubic polynomials over ℂ, with a marked critical point. Let \mathscrS_k,n be the set of all points in M₃ for which the marked critical point is strictly (k,n)-preperiodic. Milnor conjectured that the affine algebraic curves \mathscrS_k,n are irreducible, for all k ≥ 0, n>0. In this article, we show the irreducibility of eventually 2-periodic curves, i.e. \mathscrS_k,2, k≥ 0 curves. We also note that the curves, \mathscrS_k,2, k≥ 0, exhibit a possible splitting-merging phenomenon that has not been observed in earlier studies of \mathscrS_k,n curves. Finally, using the irreducibility of \mathscrS_k,2 curves, we give a new and short proof of Galois conjugacy of unicritical points lying on \mathscrS_k,2, for even natural number k.

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• The non-unit conjecture for Misiurewicz parameters

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