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Periods of Iterations of Functions with Restricted Preimage Sizes
ACM Transactions on Algorithms ( IF 0.9 ) Pub Date : 2020-06-07 , DOI: 10.1145/3378570 Rodrigo S. V. Martins 1 , Daniel Panario 2 , Claudio Qureshi 3 , Eric Schmutz 4
ACM Transactions on Algorithms ( IF 0.9 ) Pub Date : 2020-06-07 , DOI: 10.1145/3378570 Rodrigo S. V. Martins 1 , Daniel Panario 2 , Claudio Qureshi 3 , Eric Schmutz 4
Affiliation
Let [ n { = {1, …, n } and let Ω n be the set of all mappings from [ n { to itself. Let f be a random uniform element of Ω n and let T( f ) and B( f ) denote, respectively, the least common multiple and the product of the length of the cycles of f . Harris proved in 1973 that T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on n nodes. We obtain analogous results for random uniform mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k ( r ) ≥ 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form x k + a over the integers modulo p , with p ≡ 1 (mod k). We exhibit and discuss our numerical results on this heuristic.
中文翻译:
具有限制原像大小的函数的迭代周期
让 [n { = {1, ...,n } 并让 Ωn 是来自 [ 的所有映射的集合n {对自己。让F 是 Ω 的随机均匀元素n 并让 T(F ) 和 B(F ) 分别表示最小公倍数和循环长度的乘积F . 哈里斯在 1973 年证明 T 在分布中收敛到标准正态分布,并且在 2011 年,施穆茨在所有映射上获得了 T 和 B 期望的对数的渐近估计n 节点。我们获得了随机均匀映射的类似结果n =氪 原像大小限制为一组 {0,k} 的节点,其中ķ =ķ (r ) ≥ 2。这是由于使用这些映射类作为启发式模型来统计以下形式的多项式Xķ + a 在整数模上p , 和p ≡ 1 (mod k)。我们展示并讨论了我们在这个启发式上的数值结果。
更新日期:2020-06-07
中文翻译:
具有限制原像大小的函数的迭代周期
让 [