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Factoring polynomials with rational coefficients
 MATH. ANN
, 1982
"... In this paper we present a polynomialtime algorithm to solve the following problem: given a nonzero 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 polynomia ..."
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Cited by 982 (11 self)
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In this paper we present a polynomialtime algorithm to solve the following problem: given a nonzero 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 padic 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
Lifting and recombination techniques for absolute factorization
 J. Complexity
, 2007
"... Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic ..."
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Cited by 23 (7 self)
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Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.
Factorization of Polynomials
 Computing, Suppl. 4
, 1982
"... Algorithms for factoring polynomials in one or more variables over various coefficient domains are discussed. Special emphasis is given to finite fields, the integers, or algebraic extensions of the rationals, and to multivariate polynomials with integral coefficients. In particular, various squaref ..."
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Cited by 6 (0 self)
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Algorithms for factoring polynomials in one or more variables over various coefficient domains are discussed. Special emphasis is given to finite fields, the integers, or algebraic extensions of the rationals, and to multivariate polynomials with integral coefficients. In particular, various squarefree decomposition algorithms and Hensel lifting techniques are analyzed. An attempt is made to establish a complete historic trace for today's methods. The exponential worst case complexity nature of these algorithms receives attention. _______________ Appears in Computer Algebra, second edition, B. Buchberger, R. Loos, G. Collins, editors, Springer Verlag, Vienna, Austria, pp. 9511 (1982).  2  1. Introduction The problem of factoring polynomials has a long and distinguished history. D. Knuth traces the first attempts back to Isaac Newton's Arithmetica Universalis (1707) and to the astronomer Friedrich T. v. Schubert who in 1793 presented a finite step algorithm to compute the factors...
Factoring Polynomials with Rational Coefficients
, 1998
"... this paper is to present, in greater detail, this algorithm for factoring polynomials over Q. It is based on the original paper of Lenstra, Lenstra, and Lov'asz [4] of the same title and on courses in algebraic number theory and the geometry of numbers, taught by Cameron Stewart at the Universi ..."
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this paper is to present, in greater detail, this algorithm for factoring polynomials over Q. It is based on the original paper of Lenstra, Lenstra, and Lov'asz [4] of the same title and on courses in algebraic number theory and the geometry of numbers, taught by Cameron Stewart at the University of Waterloo. 2 Motivation For notational purposes, for any f ffl Z[x], let (f mod m) denote the polynomial in Z=mZ[x]whose coefficients are the respective coefficients of f reduced modulo m. In order for the algorithm to run correctly, we must choose choose our values of p and k very carefully so that both Berlekamp's algorithm and the application of Hensel's lemma will output a polynomial which will be of use to the L 3  algorithm and the factorization techniques we will use. The choice of a prime p is explained in section 3 while our choice of k is explained in section 6. This section will demonstrate the criteria given by Lenstra, Lenstra, and Lov'asz [4] for setting up the L 3 algorithm. 1 Let f ffl Z[x] be a primitive polynomial of degree n ? 0. In using Berlekamp's algorithm and Hensel's lemma, we would like to produce a polynomial h ffl Z[x] with the following properties: h has leading coefficient 1 (1) (h mod p k ) divides (f mod p k ) in Z=p k Z[x] (2) (h mod p) is irreducible in F p [x] (3) (h mod p) 2 does not divide(f mod p) in F p [x] (4) Let l = deg(h). Hence 0 ! l n. The reason for finding such an h is shown in the following proposition. Proposition 1 f has an irreducible factor h 0 in Z[x], unique up to sign, for which (h mod p) divides (h 0 mod p). Further, if g ffl Z[x] divides f, then the following are equivalent: i) (h mod p) divides (g mod p) in F p [x] ii) (h mod p k ) divides (g mod p k ) in Z=p k Z[x] iii) h 0 divides g i...
Complexity Issues in Bivariate Polynomial Factorization ABSTRACT
"... Many polynomial factorization algorithms rely on Hensel lifting and factor recombination. For bivariate polynomials we show that lifting the factors up to a precision linear in the total degree of the polynomial to be factored is sufficient to deduce the recombination by linear algebra, using trace ..."
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Many polynomial factorization algorithms rely on Hensel lifting and factor recombination. For bivariate polynomials we show that lifting the factors up to a precision linear in the total degree of the polynomial to be factored is sufficient to deduce the recombination by linear algebra, using trace recombination. Then, the total cost of the lifting and the recombination stage is subquadratic in the size of the dense representation of the input polynomial. Lifting is often the practical bottleneck of this method: we propose an algorithm based on a faster multimoduli computation for univariate polynomials and show that it saves a constant factor compared to the classical multifactor lifting algorithm.
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"... Abstract. We want to achieve efficiency for the exact computation of the dot product of two vectors over word size finite fields. We therefore compare the practical behaviors of a wide range of implementation techniques using different representations. The techniques used include floating point repr ..."
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Abstract. We want to achieve efficiency for the exact computation of the dot product of two vectors over word size finite fields. We therefore compare the practical behaviors of a wide range of implementation techniques using different representations. The techniques used include floating point representations, discrete logarithms, tabulations, Montgomery reduction, delayed modulus. Our implementations have many symbolic linear algebra applications: matrix multiplication, symbolic triangularization, system solving, exact determinant computation, matrix normal form are such examples. 1