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Visible points on curves over finite fields

Shparlinski I. E., Voloch J. F.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 3

193-199

For a prime p and an absolutely irreducible modulo p polynomial ƒ(U, V) L 1\U, V] we obtain an asymptotic formula for the number of solutions to the congruence ƒ(x,y) = a (modp) 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.