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## Factoring polynomials with rational coefficients (1982)

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Venue: | MATH. ANN |

Citations: | 955 - 11 self |

### Citations

2760 |
The art of computer programming
- Knuth
- 1998
(Show Context)
Citation Context ... < y < 1, then the powers of 2 appearing in (1.7), (1.8) and (1.9) must be replaced by the same powers of 4/(4y- 1). Remark. From (1.8) we see that a reduced basis is also reduced in the sense of [9, =-=(7)-=--I. Proof of (1.6). From (1.5) and (1.4) we see that for 1 < i < n, so by induction From (1.2) and (1.4) we now obtain It follows that for 1-<j<i<n. This proves (1.7). From (1.1), (1.2) it follows tha... |

1250 | An Introduction to the Theory of Numbers - Hardy, Wright - 1979 |

433 |
Approximate formulas for some functions of prime numbers
- Rosser, Schoenfeld
- 1962
(Show Context)
Citation Context ... R(f, f') we have (3.7) I-I q < [R(f f')[. q < p, q prime It is not difficult to prove that there is a positive constant A such that (3.8) I-I q >eAp q< p, qprimc for all p>2, see [6, Sect. 22.2]; by =-=[12]-=- we can take A=0.84 for p>101. From Hadamard's inequality (1.10) we easily obtain [R(f, f')[ < nnlf[ 2n-1 . Combining this with (3.7) and (3.8) we conclude that (3.9) p < (n log n + (2n- 1) log Ifl)/A... |

313 | Integer programming with a fixed number of variables
- LENSTRA
- 1983
(Show Context)
Citation Context ... (h modp) divides (g modp). Suppose that this is not the case. Then by (2.3) we have (2.9) ~-3 h + #3g = 1 - pv 3 for certain 2 3, #3, v3E 7/IX]. We shall derive a contradiction from this. Put e = deg=-=(9)-=- and m' = deg(b). Clearly 0 < e < m' < m. We define M = {).f + #b : 2, #~ 7/IX], deg(2) < m' - e, deg(#) < n- e} CZ+7/'X +... +7/.X "+"'- ~- 1. Let M' be the projection of M on 2g .X~+ 7/.X e+ 1 +... ... |

239 | An introduction to the geometry of numbers - Cassels - 1997 |

51 |
Multi-dimensional Continued Fraction Algorithms
- Brentjes
- 1981
(Show Context)
Citation Context ...Z (l<i<n) for l<j<t. For fixed j, let i(j) i=1 denote the largest i for which rij.0. Then we have, by the proof of (1.i1) (1.13) Ix j[ 2 ~ Ibm(j)[ 2 for 1 =<j =< t. Renumber the xj such that i(1)_-< i=-=(2)-=-=<... _<_ i(t). We claim that j____ i(j) for 1 =< j-_< t. If not, then x 1, x2, ..., xj would all belong to P,b i + Rb2 +... + IRbj_ 1, a contradiction with the linear independence of x 1, x2 ..... x ... |

50 | An inequality about factors of polynomials - Mignotte - 1974 |

50 | On Hensel factorization I - Zassenhaus - 1969 |

35 |
Generalization of the Euclidean Algorithm for Real Numbers to All Dimensions Higher Than Two
- Ferguson, Forcade
- 1979
(Show Context)
Citation Context ...nalysis of our algorithm for factoring polynomials. It may be expected that other irreducibility tests and factoring methods that depend on diophantine approximation (Cantor [3], Ferguson and Forcade =-=[5]-=-, Brentjes [2, Sect. 4A], and Zassenhaus [16]) can also be made into polynomialtime algorithms with the help of the basis reduction algorithm presented in Sect. 1. Splitting an arbitrary non-zero poly... |

18 |
A sublinear additive sieve for finding prime numbers
- Pritchard
- 1981
(Show Context)
Citation Context ...nd left to the reader. We consider the second step. Write P for the right hand side of (3.9). Then p can be found with O(P) arithmetic operations on integers of binary length O(P); here one can apply =-=[11]-=- to generate a table of prime numbers < P, or alternatively use a table ofsquarefree numbers, which is easier to generate. From p < P it also follows that Berlekamp's algorithm satisfies the estimates... |

10 |
Factorization of polynomials
- Lenstra
(Show Context)
Citation Context ...omials 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... |

7 | A remark on the Hensel factorization method - Zassenhaus - 1978 |

3 |
Irreducible polynomials with integral coefficients have succinct certificates
- Cantor
- 1981
(Show Context)
Citation Context ... the description and the analysis of our algorithm for factoring polynomials. It may be expected that other irreducibility tests and factoring methods that depend on diophantine approximation (Cantor =-=[3]-=-, Ferguson and Forcade [5], Brentjes [2, Sect. 4A], and Zassenhaus [16]) can also be made into polynomialtime algorithms with the help of the basis reduction algorithm presented in Sect. 1. Splitting ... |

1 |
Irreducibility testing and factorization of polynomials, to appear. Extended abstract
- Adleman, Odlyzko
- 1981
(Show Context)
Citation Context ...deduce from our main result that the problem of factoring such a polynomial is polynomial-time reducible to the problem of factoring positive integers. The same fact was proved by Adleman and Odlyzko =-=[1]-=- under the assumption of several deep and unproved hypotheses from number theory. The generalization of our result to algebraic number fields and to polynomials in several variables is the subject of ... |

1 | The Hensel lemma in algebraic manipulation. Cambridge - Yun - 1974 |

1 |
A new polynomial factorization algorithm (unpublished manuscript
- Zassenhaus
- 1981
(Show Context)
Citation Context ...omials. It may be expected that other irreducibility tests and factoring methods that depend on diophantine approximation (Cantor [3], Ferguson and Forcade [5], Brentjes [2, Sect. 4A], and Zassenhaus =-=[16]-=-) can also be made into polynomialtime algorithms with the help of the basis reduction algorithm presented in Sect. 1. Splitting an arbitrary non-zero polynomial feZ[X] into its content and its primit... |