License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2019.15
URN: urn:nbn:de:0030-drops-108373
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10837/
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### Counting Basic-Irreducible Factors Mod p^k in Deterministic Poly-Time and p-Adic Applications

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### 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.

### BibTeX - Entry

@InProceedings{dwivedi_et_al:LIPIcs:2019:10837,
author =	{Ashish Dwivedi and Rajat Mittal and Nitin Saxena},
title =	{{Counting Basic-Irreducible Factors Mod p^k in Deterministic Poly-Time and p-Adic Applications}},
booktitle =	{34th Computational Complexity Conference (CCC 2019)},
pages =	{15:1--15:29},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-116-0},
ISSN =	{1868-8969},
year =	{2019},
volume =	{137},
editor =	{Amir Shpilka},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10837},
URN =		{urn:nbn:de:0030-drops-108373},
doi =		{10.4230/LIPIcs.CCC.2019.15},
annote =	{Keywords: deterministic, root, counting, modulo, prime-power, tree, basic irreducible, unramified}
}

 Keywords: deterministic, root, counting, modulo, prime-power, tree, basic irreducible, unramified Collection: 34th Computational Complexity Conference (CCC 2019) Issue Date: 2019 Date of publication: 16.07.2019

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