On Algorithms to Find p-ordering

Abstract

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 a set of size n in time $\widetilde{\mathcal {O}}(nk\log p)$, where set is considered modulo $p^k$. The subsets modulo $p^k$ can be represented concisely using the notion of representative roots (Panayi, PhD Thesis, 1995; Dwivedi et al., ISSAC, 2019); a natural question is, 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 $\widetilde{\mathcal {O}}(d^2k\log p + nk \log p + nd)$, where d is the size of the succinct representation and n is the required length of the p-ordering. Another contribution 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 before (Dearden and Metzger, Eur. J. Comb., 1997; Maulick, J. Comb. Theory, Ser. A, 2001), we explicitly describe all the root sets for $k\le 4$.

The full version is available at https://arxiv.org/abs/2011.10978.