On the Convex Closure of the Graph of Modular Inversions

TL;DR

This paper investigates the convex closure of the modular inversion graph, providing bounds, heuristics, and numerical comparisons, revealing peculiarities and differences when analyzing polynomial graphs over finite rings.

Contribution

It introduces bounds, heuristics, and algorithms for analyzing the convex closure of modular inversion graphs and compares numerical results with theoretical estimates.

Findings

Numeric results differ from heuristic estimates for the inversion graph.

Peculiarities in the inversion graph are heuristically explained.

Polynomial graphs over inite rings align better with heuristics.

Abstract

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 $$ G_n=\left\{(a,b) : a,b\in \Z, ab \equiv 1 \pmod{n}, 1\leq a,b\leq n-1\right\}. $$ The heuristic is based on an asymptotic formula of R\'{e}nyi and Sulanke. After describing two algorithms to determine the convex closure, we compare the numeric results with the heuristic estimate. The numeric results do not agree with the heuristic estimate -- 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 $\mathbb{Z}_n$. In this case the numeric results are in much closer agreement with the heuristic, which strongly suggests that the the curve $xy=1\pmod{n}$ is ``atypical''.