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Counting Basic-Irreducible Factors Mod p^k in Deterministic Poly-Time and p-Adic Applications

Authors Ashish Dwivedi, Rajat Mittal, Nitin Saxena

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Author Details

Ashish Dwivedi
  • CSE, Indian Institute of Technology, Kanpur, India
Rajat Mittal
  • CSE, Indian Institute of Technology, Kanpur, India
Nitin Saxena
  • CSE, Indian Institute of Technology, Kanpur, India

Acknowledgements

We thank Vishwas Bhargava for introducing us to the open problem of factoring f mod p^3 and the related prime-power questions. A.D. thanks Sumanta Ghosh for the discussions. N.S. thanks the funding support from DST (DST/SJF/MSA-01/2013-14). R.M. would like to thank support from DST through grant DST/INSPIRE/04/2014/001799. We thank anonymous reviewers for their helpful comments and suggestions to improve the introduction section of this paper.

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Ashish Dwivedi, Rajat Mittal, and Nitin Saxena. Counting Basic-Irreducible Factors Mod p^k in Deterministic Poly-Time and p-Adic Applications. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 15:1-15:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.CCC.2019.15

Abstract

Finding an irreducible factor, of a polynomial f(x) modulo a prime p, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that counts the number of irreducible factors of f mod p. We can ask the same question modulo prime-powers p^k. The irreducible factors of f mod p^k blow up exponentially in number; making it hard to describe them. Can we count those irreducible factors mod p^k that remain irreducible mod p? These are called basic-irreducible. A simple example is in f=x^2+px mod p^2; it has p many basic-irreducible factors. Also note that, x^2+p mod p^2 is irreducible but not basic-irreducible!
We give an algorithm to count the number of basic-irreducible factors of f mod p^k in deterministic poly(deg(f),k log p)-time. This solves the open questions posed in (Cheng et al, ANTS'18 & Kopp et al, Math.Comp.'19). In particular, we are counting roots mod p^k; which gives the first deterministic poly-time algorithm to compute Igusa zeta function of f. Also, our algorithm efficiently partitions the set of all basic-irreducible factors (possibly exponential) into merely deg(f)-many disjoint sets, using a compact tree data structure and split ideals.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Representation of mathematical objects
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Algebraic complexity theory
  • Computing methodologies → Hybrid symbolic-numeric methods
  • Mathematics of computing → Combinatoric problems
  • Computing methodologies → Number theory algorithms
Keywords
  • deterministic
  • root
  • counting
  • modulo
  • prime-power
  • tree
  • basic irreducible
  • unramified

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