Improved dense multivariate polynomial factorization algorithms

Journal of Symbolic Computation

Volumn 42, Issue 4, 2007, Pages 477-494

  Lecerf, Grégoire ^a  

^a UNIVERSITÉ DE VERSAILLES SAINT QUENTIN EN YVELINES   (France)

Author keywords

Bertini's irreducibility theorem; Hensel lifting; Polynomial factorization

Indexed keywords

EID: 34047121543     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jsc.2007.01.003     Document Type: Article

References (49)

1. Bajaj C., Canny J., Garrity T., and Warren J. Factoring rational polynomials over the complex numbers. SIAM J. Comput. 22 2 (1993) 318-331
2. Bostan A., Lecerf G., Salvy B., Schost É., and Wiebelt B. Complexity issues in bivariate polynomial factorization. Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation (2004), ACM Press 42-49
3. Bürgisser P., Clausen M., and Shokrollahi M.A. Algebraic Complexity Theory (1997), Springer-Verlag
4. Chen, L., Eberly, W., Kaltofen, E., Saunders, B.D., Turner, W.J., Villard, G., 2002. Efficient matrix preconditioners for black box linear algebra. In: Structured and Infinite Systems of Linear Equations, Linear Algebra Appl., vol. 343-344, pp. 119-146 (special issue)
5. Chèze, G., 2004. Des méthodes symboliques-numériques et exactes pour la factorisation absolue des polynômes en deux variables. Ph.D. Thesis, Université de Nice-Sophia Antipolis, France
6. Chèze G., and Galligo A. Four lectures on polynomial absolute factorization. In: Dickenstein A., and Emiris I.Z. (Eds). Solving Polynomial Equations: Foundations, Algorithms, and Applications. Algorithms Comput. Math. vol. 14 (2005), Springer-Verlag 339-392
7. Chèze, G., Lecerf, G., 2005. Lifting and recombination techniques for absolute factorization, Manuscript
8. Gao S. Absolute irreducibility of polynomials via Newton polytopes. J. Algebra 237 2 (2001) 501-520
9. Gao S. Factoring multivariate polynomials via partial differential equations. Math. Comp. 72 (2003) 801-822
10. Gao S., Kaltofen E., and Lauder A. Deterministic distinct degree factorization for polynomials over finite fields. J. Symbolic Comput. 38 6 (2004) 1461-1470
11. von zur Gathen J. Irreducibility of multivariate polynomials. J. Comput. System Sci. 31 2 (1985) 225-264
12. von zur Gathen J., and Gerhard J. Modern Computer Algebra. second ed (2003), Cambridge University Press
13. von zur Gathen J., and Kaltofen E. Factoring sparse multivariate polynomials. J. Comput. System Sci. 31 (1985) 265-287
14. von zur Gathen J., and Kaltofen E. Factorization of multivariate polynomials over finite fields. Math. Comp. 45 171 (1985) 251-261
15. Heintz J., and Sieveking M. Absolute primality of polynomials is decidable in random polynomial time in the number of variables. Automata, languages and programming (Akko, 1981). Lecture Notes in Comput. Sci. vol. 115 (1981), Springer-Verlag 16-28
16. Hilbert D. Ueber die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Coefficienten. J. Reine Angew. Math. 110 (1892)
17. van der Hoeven J. The truncated Fourier transform and applications. Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation (2004), ACM Press 290-296
18. van der Hoeven, J., 2005. Notes on the truncated Fourier transform, Manuscript
19. Jouanolou J.-P. Théorèmes de Bertini et applications. Progress in Mathematics vol. 42 (1983), Birkhäuser
20. Kaltofen E. Polynomial factorization. In: Buchberger B., Collins G., and Loos R. (Eds). Computer Algebra (1982), Springer-Verlag 95-113
21. Kaltofen E. A polynomial reduction from multivariate to bivariate integral polynomial factorization. Proceedings of the 14th Symposium on Theory of Computing (1982), ACM 261-266
22. Kaltofen E. A polynomial-time reduction from bivariate to univariate integral polynomial factorization. Proceedings of the 23rd Symposium on Foundations of Computer Science (1982), IEEE 57-64
23. Kaltofen E. Effective Hilbert irreducibility. Inform. Control 66 3 (1985) 123-137
24. Kaltofen E. Fast parallel absolute irreducibility testing. J. Symbolic Comput. 1 1 (1985) 57-67
25. Kaltofen E. Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization. SIAM J. Comput. 14 2 (1985) 469-489
26. Kaltofen E. Sparse Hensel lifting. Proceedings of EUROCAL '85, vol. 2. Linz, 1985. Lecture Notes in Comput. Sci. vol. 204 (1985), Springer-Verlag 4-17
27. Kaltofen E. Deterministic irreducibility testing of polynomials over large finite fields. J. Symbolic Comput. 4 1 (1987) 77-82
28. Kaltofen E. Factorization of polynomials given by straight-line programs. In: Micali S. (Ed). Randomness and Computation. Advances in Computing Research vol. 5 (1989), JAI Press Inc. 375-412
29. Kaltofen E. Polynomial factorization 1982-1986. Computers in Mathematics, Stanford, CA, 1986. Lecture Notes in Pure and Appl. Math. vol. 125 (1990), Dekker 285-309
30. Kaltofen E. Polynomial factorization 1987-1991. LATIN '92, São Paulo, 1992. Lecture Notes in Comput. Sci. vol. 583 (1992), Springer-Verlag 294-313
31. Kaltofen E. Effective Noether irreducibility forms and applications. J. Comput. System Sci. 50 2 (1995) 274-295
32. Kaltofen E. Polynomial factorization: A success story. Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation (2003), ACM Press 3-4
33. Kaltofen E., and Saunders B.D. On Wiedemann's method of solving sparse linear systems. In: Mattson H.F., Mora T., and Rao T.R.N. (Eds). Proceedings of AAECC-9. Lect. Notes Comput. Sci. vol. 539 (1991), Springer-Verlag 29-38
34. Kaltofen E., and Trager B. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. J. Symbolic Comput. 9 3 (1990) 301-320
35. Kleiman S.L. Bertini and his two fundamental theorems. Rend. Circ. Mat. Palermo 2 Suppl. 55 (1998) 9-37 Studies in the History of Modern Mathematics, III
36. Lecerf G. Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers. J. Complexity 19 4 (2003) 564-596
37. Lecerf G. Sharp precision in Hensel lifting for bivariate polynomial factorization. Math. Comp. 75 (2006) 921-933
38. Lecerf G., and Schost É. Fast multivariate power series multiplication in characteristic zero. SADIO Electron. J. Inf. Oper. Res. 5 1 (2003) 1-10
39. Mumford D. Algebraic geometry. I Complex projective varieties. Classics in Mathematics (1995), Springer-Verlag Reprint of the 1976 edition
40. Ruppert W.M. Reduzibilität ebener Kurven. J. Reine Angew. Math. 369 (1986) 167-191
41. Ruppert W.M. Reducibility of polynomials f (x, y) modulo p. J. Number Theory 77 1 (1999) 62-70
42. Schinzel A. Polynomials with special regard to reducibility. Encyclopedia of Mathematics and its Applications vol. 77 (2000), Cambridge University Press
43. Schost É. Multivariate power series multiplication. Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation (2005), ACM Press 293-300
44. Schwartz J.T. Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27 4 (1980) 701-717
45. Shafarevich I.R. Basic Algebraic Geometry. 1 Varieties in Projective Space. second ed (1994), Springer-Verlag
46. Storjohann, A., 2000. Algorithms for matrix canonical forms. Ph.D. Thesis, ETH, Zürich, Switzerland
47. Zassenhaus H. On Hensel factorization I. J. Number Theory 1 1 (1969) 291-311
48. Zippel R. Probabilistic algorithms for sparse polynomials. Proceedings of EUROSAM '79. Lecture Notes in Comput. Sci. No. 72 (1979), Springer-Verlag 216-226
49. Zippel R. Effective Polynomial Computation (1993), Kluwer Academic Publishers