2016/06/30 by T. D. Browning, Alex Gorodnik, A. Gorodnik
Mathematics · #Advanced Combinatorial Mathematics #Affine transformation #Algebraic Geometry and Number Theory #Combinatorics #Discrete mathematics #Homogeneous #Hypersurface #Integer (computer science) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Polynomial #Pure mathematics #Quadric #Variety (cybernetics) #math.DS #math.NT #msc:11D09 #msc:11D45 #msc:11N32 #msc:20G30
paper · pdf · doi:10.1112/plms.12030
47 pages; accepted version
arxiv created 2017/02/01 · openalex publication_date 2017/03/10 · arxiv updated 2017/06/14 · openalex created_date 2021/02/01 · openalex updated_date 2026/08/05
Given a symmetric variety Y defined over Q and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski dense set of integral points on Y. We also establish an asymptotic counting formula for this set. In the special case that Y is a quadric hypersurface, we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics defined over Q.