Visible Points on Curves over Finite Fields

Abstract

For a prime $p$ and an absolutely irreducible modulo $p$ polynomial $f(U,V)\in \mathbb{Z}[U,V]$ we obtain an asymptotic formulas for the number of solutions to the congruence $f(x,y)\equiv a(\mathrm{mod}p)$ 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 $a$.