Experimental Mathematics

On the Convex Closure of the Graph of Modular Inversions

Mizan Khan, Igor E. Shparlinski, and Christian L. Yankov

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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.''

Article information

Experiment. Math., Volume 17, Issue 1 (2008), 91-104.

First available in Project Euclid: 18 November 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11A07: Congruences; primitive roots; residue systems 11H06: Lattices and convex bodies [See also 11P21, 52C05, 52C07] 11K38: Irregularities of distribution, discrepancy [See also 11Nxx] 11N25: Distribution of integers with specified multiplicative constraints

modular inversion convex hull distribution of divisors


Khan, Mizan; Shparlinski, Igor E.; Yankov, Christian L. On the Convex Closure of the Graph of Modular Inversions. Experiment. Math. 17 (2008), no. 1, 91--104. https://projecteuclid.org/euclid.em/1227031900

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