## Fast Parallel Absolute Irreducibility Testing (1985)

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Venue: | J. Symbolic Comput |

Citations: | 32 - 7 self |

### BibTeX

@ARTICLE{Kaltofen85fastparallel,

author = {Erich Kaltofen},

title = {Fast Parallel Absolute Irreducibility Testing},

journal = {J. Symbolic Comput},

year = {1985},

volume = {1},

pages = {57--67}

}

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### Abstract

We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC of Boolean circuits of polynomial size and logarithmic depth. Therefore it also belongs to the class of sequentially polynomial-time problems. Our algorithm can be extended to compute in parallel one irreducible complex factor of a multivariate integral polynomial. However, the coefficients of the computed factor are only represented modulo a not necessarily irreducible polynomial specifying a splitting field. A consequence of our algoithm is that multivariate polynomials over finite fields can be tested for absolute irreducibility in deterministic sequential polynomial time in the size of the input. We also obtain a sharp bound for the last prime p for which, when taking an absolutely irreducible integral polynomial modulo p, the polynomial's irreducibility in the algebraic closure of the finite field order p is not preserved.

### Citations

741 | Factoring polynomials with rational coefficients - Lenstra, jr, et al. - 1982 |

62 | Parallel computation for well-endowed rings and space bounded probabilistic machines - Borodin, Cook, et al. - 1983 |

43 | Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization - Kaltofen - 1985 |

41 | Fast parallel matrix and GCD computations - GATHEN, J, et al. - 1982 |

33 | Towards a Complexity Theory of Synchronous Parallel Systems. LEnseignement Nfathematique, Reveu Internationale, Geneva - Cook - 1981 |

29 | Computing in algebraic extensions - Loos - 1982 |

27 | Logarithmic depth circuits for algebraic functions - Reif - 1986 |

26 | Elements of Algebra and Algebraic Computing - Lipson - 1981 |

17 | Absolute Primality of Polynomials is Decidable in Random Polynomial Time - Heintz, Sieveking - 1981 |

17 | A note on the parallel complexity of computing the rank of order n matrices - Ibarra, Moran, et al. - 1980 |

13 | A polynomial-time reduction from bivariate to univariate integral polynomial factorization
- Kaltofen
- 1982
(Show Context)
Citation Context ...been proposed within th ramework of sequential polynomial-time complexity. For the coefficients being rational ( numbers, the first solutions are due to Lenstra et al. (1982) in the univariate and to =-=Kaltofen 1982-=-, 1983) in the dense multivariate case. It seems natural to ask whether any of these l a algorithms can be converted to a parallel one. Unfortunately, for rationals as coefficients, al lgorithms devel... |

11 | Effective Hilbert Irreducibility
- Kaltofen
- 1985
(Show Context)
Citation Context ...e an effective version of the Hil ert Irreducibility Theorem, which was the approach by Heintz, Sieveking (1981). Other ( effective versions of this theorem can be found in von zur Gathen (1983b) and =-=Kaltofen 1984-=-). The result is a random parallel algorithm which runs in (logs+ log deg( f ) + log v + log log # f ) steps wheresis the number of monomials of the input polynomial f and v O (1) the number of variab... |

9 | Factoring Sparse Multivariate Polynomials - Gathen, J - 1985 |

7 | Parallel algorithms for algebraic problems - GATHEN - 1983 |

6 | Factorization over finitely generated fields - Davenport, Trager - 1981 |

3 | Absolute primality of polynomials is decidable - Heintz, Sieveking - 1981 |

3 | Zur arithmetischen theorie der algebraischen grossen. Gottinger Nachrichten - Ostrowski - 1919 |

2 | Computations on curves - Dicrescenzo, Duval - 1984 |

2 | Ein algebraisches Kriterium f"ur absolute Irreduzibilit"at - Noether - 1922 |

2 | Ein algebraisches Kriterium f ürabsolute Irreduzibilit ät - Noether - 1922 |

1 |
Fast parallel matrix and GCD computaB tions
- orodin, Gathen, et al.
- 1982
(Show Context)
Citation Context ...be tested for absolut rreducibility we do not lose generality by working over GF(q ). Secondly, singular linear m . ( systems over finite fields can only be solved probabilistically in parallel (cf. B=-=orodin et al 1982-=-)). That means, that the algorithm might fail to produce any decision, but that with e f diminishing probability. However, we can return to the sequential technique and thus get th ollowing interestin... |

1 | Towards a complexity theory of synchronous parallel computation. D L'Enseignement mathematique 27 - ook - 1981 |

1 | A polynomial-time algorithm for factoring multivari - Gathen, Kaltofen, et al. - 1983 |

1 | A note on the parallel complexity of comput ng the rank of order n matrices - Ibarra, Moran, et al. - 1980 |

1 | Logarithmic depth circuits for algebraic functions - eif - 1983 |

1 | Apolynomial-time algorithm for factoring multivariate polynomials over finite fields - Gathen, Kaltofen, et al. - 1983 |

1 | Computing in algebraic extensions - -17Loos - 1982 |