Open Access
2008 On the Convex Closure of the Graph of Modular Inversions
Mizan Khan, Igor E. Shparlinski, Christian L. Yankov
Experiment. Math. 17(1): 91-104 (2008).


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 $ G_n={((a,b) : a,b\in \Z,\; ab \equiv 1$ (mod $n$), $1\leq a,b\leq n-1}$. 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 $\Z_n$. 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.''


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Mizan Khan. Igor E. Shparlinski. Christian L. Yankov. "On the Convex Closure of the Graph of Modular Inversions." Experiment. Math. 17 (1) 91 - 104, 2008.


Published: 2008
First available in Project Euclid: 18 November 2008

zbMATH: 1234.11005
MathSciNet: MR2410119

Primary: 11A07 , 11H06 , 11K38 , 11N25

Keywords: Convex hull , distribution of divisors , modular inversion

Rights: Copyright © 2008 A K Peters, Ltd.

Vol.17 • No. 1 • 2008
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