关于模逆图的凸闭包
On the Convex Closure of the Graph of Modular Inversions
摘要：
本文给出了集合$$ G_n=\left\{(a,b) : a,b\in \Z, ab \equiv 1 \pmod{n}, 1\leq a,b\leq n-1\right\}. $$的凸闭包的顶点数的上、下界和一个启发式估计。在描述了确定凸闭包的两种算法之后，我们将数值结果与启发式估计进行了比较。数值结果不同意与启发式估计-有一些有趣的特点，我们提供了一个启发式的解释。然后，我们描述了一些数值工作的凸封闭图的随机二次和三次多项式在$\mathbb{Z}_n$。在这种情况下，数值结果与启发式更接近，这强烈表明曲线$xy=1\pmod{n}$是“渐进的”。
英文摘要：
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 Rényi 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''.
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