Visible Points on Curves over Finite Fields
Volume 55 / 2007
Bulletin Polish Acad. Sci. Math. 55 (2007), 193-199
MSC: 11A07, 11K38, 11L40.
DOI: 10.4064/ba55-3-1
Abstract
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 $a$.
