{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article12515","name":"Counting Basic-Irreducible Factors Mod p^k in Deterministic Poly-Time and p-Adic Applications","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!\r\nWe 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.","keywords":["deterministic","root","counting","modulo","prime-power","tree","basic irreducible","unramified"],"author":[{"@type":"Person","name":"Dwivedi, Ashish","givenName":"Ashish","familyName":"Dwivedi","email":"mailto:ashish@cse.iitk.ac.in","affiliation":"CSE, Indian Institute of Technology, Kanpur, India"},{"@type":"Person","name":"Mittal, Rajat","givenName":"Rajat","familyName":"Mittal","email":"mailto:rmittal@cse.iitk.ac.in","affiliation":"CSE, Indian Institute of Technology, Kanpur, India"},{"@type":"Person","name":"Saxena, Nitin","givenName":"Nitin","familyName":"Saxena","email":"mailto:nitin@cse.iitk.ac.in","affiliation":"CSE, Indian Institute of Technology, Kanpur, India"}],"position":15,"pageStart":"15:1","pageEnd":"15:29","dateCreated":"2019-07-16","datePublished":"2019-07-16","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dwivedi, Ashish","givenName":"Ashish","familyName":"Dwivedi","email":"mailto:ashish@cse.iitk.ac.in","affiliation":"CSE, Indian Institute of Technology, Kanpur, India"},{"@type":"Person","name":"Mittal, Rajat","givenName":"Rajat","familyName":"Mittal","email":"mailto:rmittal@cse.iitk.ac.in","affiliation":"CSE, Indian Institute of Technology, Kanpur, India"},{"@type":"Person","name":"Saxena, Nitin","givenName":"Nitin","familyName":"Saxena","email":"mailto:nitin@cse.iitk.ac.in","affiliation":"CSE, Indian Institute of Technology, Kanpur, India"}],"copyrightYear":"2019","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.CCC.2019.15","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/1711.01355","https:\/\/www.cse.iitk.ac.in\/users\/nitin\/papers\/factor-mod-p4.pdf","https:\/\/doi.org\/10.1090\/mcom\/3431"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6340","volumeNumber":137,"name":"34th Computational Complexity Conference (CCC 2019)","dateCreated":"2019-07-16","datePublished":"2019-07-16","editor":{"@type":"Person","name":"Shpilka, Amir","givenName":"Amir","familyName":"Shpilka","email":"mailto:shpilka@tauex.tau.ac.il","affiliation":"Tel Aviv University, Tel Aviv, 69978, Israel"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article12515","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6340"}}}