The concept of p-ordering for a prime p was introduced by Manjul Bhargava (in his PhD thesis) to develop a generalized factorial function over an arbitrary subset of integers. This notion of p-ordering provides a representation of polynomials modulo prime powers, and has been used to prove properties of roots sets modulo prime powers. We focus on the complexity of finding a p-ordering given a prime p, an exponent k and a subset of integers modulo p^k. Our first algorithm gives a p-ordering for set of size n in time O(nk\log p), where set is considered modulo p^k. The subsets modulo p^k can be represented succinctly using the notion of representative roots (Panayi, PhD Thesis, 1995; Dwivedi et.al, ISSAC, 2019); a natural question would be, can we find a p-ordering more efficiently given this succinct representation. Our second algorithm achieves precisely that, we give a p-ordering in time O(d^2k\log p + nk \log p + nd), where d is the size of the succinct representation and n is the required length of the p-ordering. Another contribution that we make is to compute the structure of roots sets for prime powers p^k, when k is small. The number of root sets have been given in the previous work (Dearden and Metzger, Eur. J. Comb., 1997; Maulick, J. Comb. Theory, Ser. A, 2001), we explicitly describe all the root sets for p^2, p^3 and p^4.

中文翻译：

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关于找到p阶的算法

Manjul Bhargava（在其博士学位论文中）引入了质数p的p排序概念，以在任意整数子集上开发广义阶乘函数。p阶的概念提供了模素数幂的多项式表示，并已被用来证明模素数幂的根集的性质。我们专注于在给定素数p，指数k和以p ^ k为模的整数子集的情况下找到p阶的复杂性。我们的第一个算法对时间为O（nk \ log p）的大小为n的集合给出p排序，其中集合被视为对p ^ k取模。可以使用代表性根的概念来简洁地表示模p ^ k的子集（Panayi，PhD Thesis，1995; Dwivedi et al。，ISSAC，2019）; 一个自然的问题是，在这种简洁的表示形式下，我们能否更有效地找到p阶。我们的第二个算法精确地实现了，我们在时间O（d ^ 2k \ log p + nk \ log p + nd）中给出p阶，其中d是简洁表示的大小，n是p所需的长度-订购。我们做出的另一个贡献是，当k较小时，计算素数幂p ^ k的根集的结构。在先前的工作中已经给出了根集的数量（Dearden和Metzger，欧洲E.J. Comb。，1997； Maulick，J.Comb.Theory，Ser.A，2001），我们明确地描述了p的所有根集。 ^ 2，p ^ 3和p ^ 4。