Visible Points on Curves over Finite Fields

Volume 55 / 2007

Igor E. Shparlinski, José Felipe Voloch Bulletin Polish Acad. Sci. Math. 55 (2007), 193-199 MSC: 11A07, 11K38, 11L40. DOI: 10.4064/ba55-3-1


For a prime $p$ and an absolutely irreducible modulo $p$ polynomial $f(U,V) \in \mathbb Z[U,V]$ we obtain an asymptotic formula for the number of solutions to the congruence $f(x,y) \equiv a \pmod p$ in positive integers $x \le X$, $y \le Y$, with the additional condition $\gcd(x,y)=1$. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over $a$ for a fixed prime $p$, and also on average over $p$ for a fixed integer $a$.


  • Igor E. ShparlinskiDepartment of Computing
    Macquarie University
    Sydney, NSW 2109, Australia
  • José Felipe VolochDepartment of Mathematics
    University of Texas
    Austin, TX 78712, U.S.A.

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