On the convex closure of the graph of modular inversions

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 C _n = {(a,b) : a,b ε ℤ, ab = 1 (mod n), 1 ≤ a,b ≤ n - 1}. The heuristic is. based on an asymptotic formula of Renyi and Su-lanke. 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."