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Sparsity of curves and additive and multiplicative expansion of rational maps over finite fields

Volume 188 / 2019

László Mérai, Igor E. Shparlinski Acta Arithmetica 188 (2019), 401-411 MSC: Primary 11D79, 11G20, 12D10, 30C15. DOI: 10.4064/aa180307-20-8 Published online: 12 April 2019


For a prime $p$ and a polynomial $F(X,Y)$ over the finite field $\mathbb{F}_p$ of $p$ elements, we give upper bounds on the number of solutions of $$ F(x,y)=0, \quad x\in\mathcal{A}, \, y\in \mathcal{B}, $$ where $\mathcal A$ and $\mathcal B$ are very small intervals or subgroups. These bounds can be considered as positive characteristic analogues of the result of Bombieri and Pila (1989) on sparsity of integral points on curves. As an application we prove that distinct consecutive elements in sequences generated by compositions of several rational functions are not contained in any short intervals or small subgroups.


  • László MéraiJohann Radon Institute for Computational
    and Applied Mathematics
    Austrian Academy of Sciences
    Institute of Financial Mathematics
    and Applied Number Theory
    Johannes Kepler University
    Altenberger Straße 69
    A-4040 Linz, Austria
  • Igor E. ShparlinskiSchool of Mathematics and Statistics
    University of New South Wales
    Sydney, NSW 2052, Australia

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