On the convex closure of the graph of modular inversions | Macquarie University ResearchOnline

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Title: On the convex closure of the graph of modular inversions
Related: Experimental mathematics, Vol. 17, Issue 1, p.91-104
Publisher: A. K. Peters
Date: 2008
FoR/RFCD Code(s): 010100 Pure Mathematics 010200 Applied Mathematics 010300 Numerical and Computational Mathematics
Author/Creator: Khan, Mizan R
Author/Creator: Shparlinski, Igor E
Author/Creator: Yankov, Christian L
Description: In this paper we give upper and lower bounds as well as a heuristic estimate on the number of vertices of the convex closure of the set Gn = {(a, b) : a, b ∈ ℤ, ab ≡ 1 (mod n) , 1 ≤ a, b ≤ n - 1} . The heuristic is based on an asymptotic formula of Renyi and Sulanke. After describing two algorithms to determine the convex closure, we compare the numeric results with the heuristic estimate, and find that they do not agree—there are some interesting peculiarities, for which we provide a heuristic explanation. We then describe some numerical work on the convex closure of the graph of random quadratic and cubic polynomials over ℤn. In this case the numeric results are in much closer agreement with the heuristic, which strongly suggests that the curve xy = 1(mod n) is "atypical."
Description: 14 page(s)
Subject Keyword: 010100 Pure Mathematics
Subject Keyword: 010200 Applied Mathematics
Subject Keyword: 010300 Numerical and Computational Mathematics
Resource Type: journal article
Organisation: Macquarie University. Dept. of Computing

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