Lifting and recombination techniques for absolute factorization

Journal of Complexity

Volumn 23, Issue 3, 2007, Pages 380-420

  Chèze, Guillaume ^a   Lecerf, Grégoire ^b  

^a UNIVERSITÉ DE NICE SOPHIA ANTIPOLIS   (France)

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

Author keywords

Absolute factorization; Absolute primality; Polynomial factorization

Indexed keywords

ALGORITHMS; POLYNOMIALS;

ABSOLUTE FACTORIZATION; BIVARIATE POLYNOMIAL; POLYNOMIAL FACTORIZATION;

FACTORIZATION;

EID: 34249705308     PISSN: 0885064X     EISSN: 10902708     Source Type: Journal    
DOI: 10.1016/j.jco.2007.01.008     Document Type: Article

References (77)

1. Abu Salem F., Gao S., and Lauder A.G.B. Factoring polynomials via polytopes. Proceedings of ISSAC 2004 (2004), ACM Press, New York 4-11
2. Bajaj C., Canny J., Garrity T., and Warren J. Factoring rational polynomials over the complex numbers. SIAM J. Comput. 22 2 (1993) 318-331
3. K. Belabas, M. van Hoeij, J. Klüners, A. Steel, Factoring polynomials over global fields, manuscript, October 2004.
4. Bini D., and Pan V.Y. Polynomial and Matrix Computations, Fundamental Algorithms, Progress in Theoretical Computer Science vol. 1 (1994), Birkhäuser, Basel
5. Bostan A., Flajolet Ph., Salvy B., and Schost E. Fast computation of special resultants. J. Symbolic Comput. 41 1 (2006) 1-29
6. Bostan A., Lecerf G., Salvy B., Schost E., and Wiebelt B. Complexity issues in bivariate polynomial factorization. Proceedings of ISSAC 2004 (2004), ACM Press, New York 42-49
7. Bostan A., Lecerf G., and Schost E. Tellegen's principle into practice. Proceedings of ISSAC 2003 (2003), ACM Press, New York 37-44
8. Bostan A., and Schost E. Polynomial evaluation and interpolation on special sets of points. J. Complexity 21 4 (2005) 420-446
9. M. Bronstein, Computer algebra algorithms for linear ordinary differential and difference equations, in: European Congress of Mathematics, vol. II, Barcelona, 2000, Progr. Math., 202 (2001), pp. 105-119.
10. Bronstein M. Symbolic integration. I Transcendental functions. second ed. (2005), Springer, Berlin
11. M. Bronstein, B.M. Trager, A reduction for regular differential systems, manuscript 2003.
12. Bürgisser P., Clausen M., and Shokrollahi M.A. Algebraic Complexity Theory (1997), Springer, Berlin
13. Chèze G. Absolute polynomial factorization in two variables and the knapsack problem. Proceedings of ISSAC 2004 (2004), ACM Press, New York 87-94
14. G.Chèze. Des méthodes symboliques-numériques et exactes pour la factorisation absolue des polynômes en deux variables, Ph.D. Thesis, Université de Nice-Sophia Antipolis, France, 2004.
15. 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 Computational Mathematics vol. 14 (2005), Springer, Berlin 339-392
16. Chèze G., and Galligo A. From an approximate to an exact absolute polynomial factorization. J. Symbolic Comput. 41 6 (2006) 682-696
17. Corless R.M., Galligo A., Kotsireas I.S., and Watt S.M. A geometric-numeric algorithm for absolute factorization of multivariate polynomials. Proceedings of ISSAC 2002 (2002), ACM Press, New York 37-45
18. Cormier O., Singer M.F., Trager B.M., and Ulmer F. Linear differential operators for polynomial equations. J. Symbolic Comput. 34 5 (2002) 355-398
19. Dellière S. On the links between triangular sets and dynamic constructible closure. J. Pure Appl. Algebra 163 1 (2001) 49-68
20. Duval D. Absolute factorization of polynomials: a geometric approach. SIAM J. Comput. 20 1 (1991) 1-21
21. Duval D. Évaluation dynamique et clôture algébrique en Axiom. J. Pure Appl. Algebra 99 (1995) 267-295
22. R. Dvornicich, C. Traverso, Newton symmetric functions and the arithmetic of algebraically closed fields, in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Menorca, 1987, Lecture Notes in Computer Science, vol. 356, Springer, Berlin, 1989, pp. 216-224.
23. A. Frühbis-Krüger, G. Pfister, Some applications of resolution of singularities from a practical point of view, in: Proceedings of the Conference Computational Commutative and Non-commutative Algebraic Geometry, Chisinau 2004, vol. 196, NATO Science Series III, Computer and Systems Sciences, 2005, pp. 104-117.
24. Galligo A., and Rupprecht D. Irreducible decomposition of curves. J. Symbolic Comput. 33 5 (2002) 661-677
25. Gao S. Absolute irreducibility of polynomials via Newton polytopes. J. Algebra 237 2 (2001) 501-520
26. Gao S. Factoring multivariate polynomials via partial differential equations. Math. Comput. 72 242 (2003) 801-822
27. S. Gao, J. von zur Gathen, Berlekamp's and Niederreiter's polynomial factorization algorithms, in: Finite Fields: Theory, Applications, and Algorithms, Las Vegas, NV, 1993, Contemp. Math., 168 (1994) 101-116.
28. Gao S., Kaltofen E., and Lauder A. Deterministic distinct degree factorization for polynomials over finite fields. J. Symbolic Comput. 38 6 (2004) 1461-1470
29. Gao S., Kaltofen E., May J., Yang Z., and Zhi L. Approximate factorization of multivariate polynomials via differential equations. Proceedings of ISSAC 2004 (2004), ACM Press, New York 167-174
30. Gao S., and Lauder A.G.B. Decomposition of polytopes and polynomials. Discrete Comput. Geom. 26 1 (2001) 89-104
31. S.Gao, A.G.B. Lauder, Fast absolute irreducibility testing via Newton polytopes, manuscript, 2004.
32. Gao S., and Rodrigues V.M. Irreducibility of polynomials modulo p via Newton polytopes. J. Number Theory 101 1 (2003) 32-47
33. von zur Gathen J., and Gerhard J. Modern Computer Algebra. second ed. (2003), Cambridge University Press, Cambridge, MA
34. von zur Gathen J., and Lücking T. Subresultants revisited. Theor. Comput. Sci. 297 1-3 (2003) 199-239
35. G.-M. Greuel, G. Pfister, H. Schönemann, Singular 3.0, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2005 〈http://www.singular.uni-kl.de〉.
36. Grigoriev D.Yu., and Chistov A.L. Fast factorization of polynomials into irreducible ones and the solution of systems of algebraic equations. Dokl. Akad. Nauk SSSR 275 6 (1984) 1302-1306
37. J. Heintz, M. Sieveking, Absolute primality of polynomials is decidable in random polynomial time in the number of variables, in: Automata, Languages and Programming, Akko, 1981, Lecture Notes in Computer Science, vol. 115, Springer, Berlin, 1981, pp. 16-28.
38. van Hoeij M. Factoring polynomials and the knapsack problem. J. Number Theory 95 2 (2002) 167-189
39. van Hoeij M., Ragot J.-F., Ulmer F., and Weil J.-A. Liouvillian solutions of linear differential equations of order three and higher. J. Symbolic Comput. 28 4-5 (1999) 589-609
40. Kaltofen E. Fast parallel absolute irreducibility testing. J. Symbolic Comput. 1 1 (1985) 57-67
41. Kaltofen E. Computing the irreducible real factors and components of an algebraic curve. Appl. Algebra Eng. Comm. Comput. 1 2 (1990) 135-148
42. E. Kaltofen, Polynomial factorization 1982-1986, in: Computers in Mathematics, Stanford, CA, 1986, Lecture Notes in Pure and Applied Mathematics, vol. 125, Dekker, New York, 1990, pp. 285-309.
43. E. Kaltofen, Polynomial factorization 1987-1991, in: LATIN '92, São Paulo, 1992, Lecture Notes in Computer Science, vol. 583, Springer, Berlin, 1992, pp. 294-313.
44. Kaltofen E. Effective Noether irreducibility forms and applications. J. Comput. System Sci. 50 2 (1995) 274-295
45. Kaltofen E. Polynomial factorization: a success story. Proceedings of ISSAC 2003 (2003), ACM Press, New York 3-4
46. 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, Lecture Notes in Computer Science vol. 539 (1991), Springer, Berlin 29-38
47. Lazard D., and Rioboo R. Integration of rational functions: rational computation of the logarithmic part. J. Symbolic Comput. 9 2 (1990) 113-115
48. Lecerf G. Sharp precision in Hensel lifting for bivariate polynomial factorization. Math. Comput. 75 (2006) 921-933
49. G. Lecerf, Improved dense multivariate polynomial factorization algorithms, J. Symbolic Comput. (2007), to appear. doi:10.1016/j.jsc.2007.01.003.
50. The Magma computational algebra system for algebra, number theory and geometry. 〈http://magma.maths.usyd.edu.au/magma/〉. Computational Algebra Group, School of Mathematics and Statistics, The University of Sydney, NSW, Australia, 2006.
51. Mignotte M., and Ştefǎnescu D. Polynomials, An Algorithmic Approach (1999), Springer, Berlin
52. Mulders T. A note on subresultants and the Lazard/Rioboo/Trager formula in rational function integration. J. Symbolic Comput. 24 1 (1997) 45-50
53. Niederreiter H. A new efficient factorization algorithm for polynomials over small finite field. Appl. Algebra Eng. Comm. Comput. 4 2 (1993) 81-87
54. Paterson M., and Stockmeyer L. On the number of nonscalar multiplications necessary to evaluate polynomials. SIAM J. on Comput. 2 1 (1973) 60-66
55. Pohst M.E. Factoring polynomials over global fields. I, J. Symbolic Comput. 39 6 (2005) 617-630
56. J.-F. Ragot, Sur la factorisation absolue des polynômes, Ph.D. Thesis, Université de Limoges, France, 1997.
57. Ragot J.-F. Probabilistic absolute irreducibility test for polynomials. J. Pure Appl. Algebra 172 1 (2002) 87-107
58. M. Rothstein, Aspects of symbolic integration and simplification of exponential and primitive functions, Ph.D. Thesis, University of Wisconsin-Madison, USA, 1976.
59. M. Rothstein, A new algorithm for the integration of exponential and logarithmic functions, in: Proceedings of the 1977 MACSYMA Users Conference, NASA Pub. CP-2012, 1977, pp. 263-274.
60. Ruppert W.M. Reduzibilität ebener Kurven. J. Reine Angew. Math. 369 (1986) 167-191
61. Ruppert W.M. Reducibility of polynomials f (x, y) modulo p. J. Number Theory 77 1 (1999) 62-70
62. Rupprecht D. Semi-numerical absolute factorization of polynomials with integer coefficients. J. Symbolic Comput. 37 (2004) 557-574
63. Sasaki T., and Sasaki M. A unified method for multivariate polynomial factorizations. Japan J. Indust. Appl. Math. 10 1 (1993) 21-39
64. Schinzel A. Polynomials with Special Regard to Reducibility, of Encyclopedia of Mathematics and its Applications vol. 77 (2000), Cambridge University Press, Cambridge, MA
65. Schwartz J.T. Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27 4 (1980) 701-717
66. Singer M.F., and Ulmer F. Linear differential equations and products of linear forms. J. Pure Appl. Algebra 117/118 (1997) 549-563
67. Sommese A.J., Verschelde J., and Wampler C.W. Symmetric functions applied to decomposing solution sets of polynomial systems. SIAM J. Numer. Anal. 40 6 (2002) 2026-2046
68. Sommese A.J., Verschelde J., and Wampler C.W. Advances in polynomial continuation for solving problems in kinematics. ASME J. Mech. Des. 126 2 (2004) 262-268
69. A. Steel, A new scheme for computing with algebraically closed fields, in: Algorithmic number theory (Sydney, 2002), vol. 2369, Springer, Berlin, 2002, pp. 491-505.
70. A. Storjohann, Algorithms for matrix canonical forms, Ph.D. Thesis, ETH, Zürich, Switzerland, 2000.
71. B.M. Trager, Algebraic factoring and rational function integration, in: Proceedings of the third ACM Symposium on Symbolic and Algebraic Computation, ACM Press, New York, 1976, pp. 219-226
72. B.M. Trager, Integration of algebraic functions, Ph.D. Thesis, MIT, USA, 1984.
73. C. Traverso, A study on algebraic algorithms: the normalization, Rend. Sem. Mat. Univ. Politec. Torino, (1985) 111-130 (special issue).
74. Zassenhaus H. On Hensel factorization I. J. Number Theory 1 1 (1969) 291-311
75. Zassenhaus H. A remark on the Hensel factorization method. Math. Comput. 32 141 (1978) 287-292
76. R. Zippel, Probabilistic algorithms for sparse polynomials, in: Proceedings EUROSAM' 79, Lecture Notes in Computer Science, vol. 72, Springer, Berlin, 1979, pp. 216-226.
77. Zippel R. Effective Polynomial Computation (1993), Kluwer Academic Publishers, Dordrecht