On pseudopoints of algebraic curves | Macquarie University ResearchOnline

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Title: On pseudopoints of algebraic curves
Related: Archiv der Mathematik, Vol. 95, No. 6 (2010), p.529-537
DOI: 10.1007/s00013-010-0200-7
Publisher: Birkhäuser Basel
Date: 2010
FoR/RFCD Code(s): 010100 Pure Mathematics
Author/Creator: Farashahi, Reza R
Author/Creator: Shparlinski, Igor E
Description: Following Kraitchik and Lehmer, we say that a positive integer n ≡ 1 (mod 8) is an x-pseudosquare if it is a quadratic residue for each odd prime p ≤ x, yet it is not a square. We extend this definition to algebraic curves and say that n is an x-pseudopoint of a curve defined by f(U, V) = 0 (where f Î mathbbZ[U, V]fZ[UV] ) if for all sufficiently large primes p ≤ x the congruence f(n, m) ≡ 0 (mod p) is satisfied for some m. We use the Bombieri bound of exponential sums along a curve to estimate the smallest x-pseudopoint, which shows the limitations of the modular approach to searching for points on curves.
Description: 9 page(s)
Subject Keyword: 010100 Pure Mathematics
Subject Keyword: algebraic curve
Subject Keyword: pseudopoint
Resource Type: journal article
Organisation: Macquarie University. Dept. of Computing

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