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Visible Points on Curves over Finite Fields

Authors
  • Shparlinski, I. E.
  • Voloch, J. F.
Type
Preprint
Publication Date
Apr 18, 2007
Submission Date
Apr 18, 2007
Identifiers
arXiv ID: 0704.2446
Source
arXiv
License
Unknown
External links

Abstract

For a prime $p$ and an absolutely irreducible modulo $p$ polynomial $f(U,V) \in \Z[U,V]$ we obtain an asymptotic formulas 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 $a$.

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