Factoring polynomials with rational coefficients

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Factoring polynomials with rational coefficients (1982)

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by A. K. Lenstra , H. W. Lenstra , L. Lovasz

Venue:	MATH. ANN
Citations:	955 - 11 self

BibTeX

@ARTICLE{Lenstra82factoringpolynomials,
    author = {A. K. Lenstra and H. W. Lenstra and L. Lovasz},
    title = {Factoring polynomials with rational coefficients},
    journal = {MATH. ANN},
    year = {1982},
    volume = {261},
    pages = {515--534}
}

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Abstract

In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomials feZ[X] into irreducible factors in Z[X]. Here we call f ~ Z[X] primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. [8]. Its running time, measured in bit operations, is O(nl2+n9(log[fD3). Here f~Tl[X] is the polynomial to be factored, n = deg(f) is the degree of f, and for a polynomial ~ a ~ i with real coefficients a i. i An outline of the algorithm is as follows. First we find, for a suitable small prime number p, a p-adic irreducible factor h of f, to a certain precision. This is done with Berlekamp's algorithm for factoring polynomials over small finite fields, combined with Hensel's lemma. Next we look for the irreducible factor h o of f in

Keyphrases

rational coefficient irreducible factor small finite field algorithm performs primitive polynomial non-zero polynomial fe log fd3 following problem bit operation p-adic irreducible factor polynomial-time algorithm common divisor nl2 n9 real coefficient running time suitable small prime number certain precision

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