Visible Points on Curves over Finite Fields

Igor E. Shparlinski ¹   ; José Felipe Voloch ²

¹ Department of Computing Macquarie University Sydney, NSW 2109, Australia
² Department of Mathematics University of Texas Austin, TX 78712, U.S.A.

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) no. 3, pp. 193-199

Résumé

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 $a$.

Détail

DOI : 10.4064/ba55-3-1

Keywords: prime absolutely irreducible modulo polynomial mathbb obtain asymptotic formula number solutions congruence equiv pmod positive integers additional condition gcd solutions have natural interpretation solutions which visible origin these formulas derived average fixed prime average fixed integer