Simultaneously preperiodic points for families of polynomials in normal\n form

2016/10/28 by Dragos Ghioca, Ghioca, Dragos, Liang-Chung Hsia +3
Mathematics · #Meromorphic and Entire Functions #Advanced Differential Equations and Dynamical Systems #Algebraic and Geometric Analysis

paper · pdf · doi:10.48550/arxiv.1610.09422

Abstract

Let d>m>1 be integers, let c₁,\…, c_m+1 be distinct complex\nnumbers, and let \f(z):=z^d+t₁z^m-1+t₂z^m-2+\⋯ +\nt_m-1z+t_m be an m-parameter family of polynomials. We prove that the set\nof m-tuples of parameters (t₁,\…, t_m)\∈\ℂ^m with the property\nthat each c_i (for i=1,\…, m+1) is preperiodic under the action of the\ncorresponding polynomial \f(z) is contained in finitely many\nhypersurfaces of the parameter space mathbbA^m.\n

Citations

• Torsion points on families of squares of elliptic curves
• Torsion points on families of products of elliptic curves

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• Common preperiodic points for quadratic polynomials
• Simultaneously preperiodic points for a family of polynomials in positive characteristic
• Simultaneously preperiodic integers for quadratic polynomials
• Uniform unlikely intersections for unicritical polynomials