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Visible Points on Curves over Finite Fields

Igor E. Shparlinski, José Felipe Voloch (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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.

Visible Points on Modular Exponential Curves

Tsz Ho Chan, Igor E. Shparlinski (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

We obtain an asymptotic formula for the number of visible points (x,y), that is, with gcd(x,y) = 1, which lie in the box [1,U] × [1,V] and also belong to the exponential modular curves y a g x ( m o d p ) . Among other tools, some recent results of additive combinatorics due to J. Bourgain and M. Z. Garaev play a crucial role in our argument.

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