For a prime p and an absolutely irreducible modulo p polynomial f(U,V) ∈ ℤ[U,V] we obtain an asymptotic formula for the number of solutions to the congruence f(x,y) ≡ a (mod p) in positive integers x ≤ X, y ≤ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-3-1,
author = {Igor E. Shparlinski and Jos\'e Felipe Voloch},
title = {Visible Points on Curves over Finite Fields},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {55},
year = {2007},
pages = {193-199},
zbl = {1156.11006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-3-1}
}
Igor E. Shparlinski; José Felipe Voloch. Visible Points on Curves over Finite Fields. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) pp. 193-199. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-3-1/