Visible Points on Curves over Finite Fields

Igor E. Shparlinski, and José Felipe Voloch. "Visible Points on Curves over Finite Fields." Bulletin of the Polish Academy of Sciences. Mathematics 55.3 (2007): 193-199. <http://eudml.org/doc/280646>.

@article{IgorE2007,
abstract = {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.},
author = {Igor E. Shparlinski, José Felipe Voloch},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {points visible from the origin, absolutely irreducible polynomial},
language = {eng},
number = {3},
pages = {193-199},
title = {Visible Points on Curves over Finite Fields},
url = {http://eudml.org/doc/280646},
volume = {55},
year = {2007},
}

TY - JOUR
AU - Igor E. Shparlinski
AU - José Felipe Voloch
TI - Visible Points on Curves over Finite Fields
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2007
VL - 55
IS - 3
SP - 193
EP - 199
AB - 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.
LA - eng
KW - points visible from the origin, absolutely irreducible polynomial
UR - http://eudml.org/doc/280646
ER -