Change of order for regular chains in positive dimension

Theoretical Computer Science

Volumn 392, Issue 1-3, 2008, Pages 37-65

  Dahan, Xavier ^a   Jin, Xin ^b   Moreno Maza, Marc ^b   Schost, Éric ^b  

^a ÉCOLE POLYTECHNIQUE   (France)

^b UNIVERSITY OF WESTERN ONTARIO   (Canada)

Author keywords

Change of order; Newton iteration; Regular chains

Indexed keywords

COMPUTER SYSTEM FIREWALLS; FORMAL METHODS; PROBLEM SOLVING; SUBROUTINES;

ALGEBRAIC VARIABLES; CHANGE OF ORDER; MODULAR METHODS; NEWTON ITERATION; REGULAR CHAINS;

ALGEBRA;

EID: 38849106829     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.tcs.2007.10.003     Document Type: Article

References (60)

1. P. Aubry, Ensembles triangulaires de polynômes et résolution de systèmes algébriques. Implantation en Axiom. Ph.D. thesis, Université Paris VI, 1999
2. Aubry P., Lazard D., and Moreno Maza M. On the theories of triangular sets. J. Symbolic Comput. 28 1-2 (1999) 45-124
3. Aubry P., and Valibouze A. Using Galois ideals for computing relative resolvents. J. Symbolic Comput. 30 6 (2000) 635-651
4. Baur W., and Strassen B. The complexity of partial derivatives. Theoret. Comput. Sci. 22 3 (1983) 317-330
5. Berkowitz S.J. On computing the determinant in small parallel time using a small number of processors. Inform. Process. Lett. 18 3 (1984) 147-150
6. D. Bernstein, Fast multiplication and its applications, Joe Buhler, Peter Stevenhagen (Eds.) in: Algorithmic Number Theory (in press) Preprint, 2003
7. Boulier F., Lemaire F., and Moreno Maza M. PARDI!. ISSAC'01 (2001), ACM Press 38-47
8. Boulier F., Lemaire F., and Moreno Maza M. Well known theorems on triangular systems and the D5 principle. Transgressive Comput. (2006) 79-92
9. Bürgisser P., Clausen M., and Shokrollahi M.A. Algebraic Complexity Theory. Grundlehren der mathematischen Wissenschaften vol. 315 (1997), Springer-Verlag
10. Busé L., and Chardin M. Implicitizing rational hypersurfaces using approximation complexes. J. Symbolic Comput. 40 4-5 (2005) 1150-1168
11. Cafure A., and Matera G. Fast computation of a rational point of a variety over a finite field. Math. Comput. 75 256 (2006) 2049-2085
12. Cafure A., and Matera G. Improved explicit estimates on the number of solutions of equations over a finite field. Finite Fields Appl. 12 2 (2006) 155-185
13. Cantor D.G., and Kaltofen E. On fast multiplication of polynomials over arbitrary algebras. Acta Inform. 28 7 (1991) 693-701
14. Collart S., Kalkbrener M., and Mall D. Converting bases with the Gröbner walk. J. Symbolic Comput. 24 3-4 (1997) 465-470
15. Cox D. Curves, surfaces, and syzygies. Topics in algebraic geometry and geometric modeling. Contemp. Math. vol. 334 (2003), Amer. Math. Soc. 131-150
16. Cox D., Little J., and O'Shea D. Using Algebraic Geometry. Graduate Text in Mathematics vol. 185 (1998), Springer-Verlag
17. Dahan X., Moreno Maza M., Schost É., Wu W., and Xie Y. Lifting techniques for triangular decompositions. ISSAC'05 (2005), ACM Press 108-115
18. Dahan X., Moreno Maza M., Schost É., and Xie Y. On the complexity of the D5 principle. Transgressive Comput. (2006) 149-168
19. Dahan X., and Schost É. Sharp estimates for triangular sets. ISSAC'04 (2004), ACM Press 103-110
20. C. D'Andrea, A. Khetan, Implicitization of rational surfaces with toric varieties. Preprint, 2005
21. Eisenbud D. Commutative Algebra. Graduate Text in Mathematics vol. 150 (1995), Springer-Verlag
22. Faugère J.-C., Gianni P., Lazard D., and Mora T. Efficient computation of zero-dimensional Gröbner bases by change of ordering. J. Symbolic Comput. 16 4 (1993) 329-344
23. Filatei A., Li X., Moreno Maza M., and Schost É. Implementation techniques for fast polynomial arithmetic in a high-level programming environment. ISSAC'06 (2006), ACM Press 93-100
24. Freeman T.S., Imirzian G., Kaltofen E., and Lakshman Y. DAGWOOD: A system for manipulating polynomials given by straight-line programs. ACM Trans. Math. Software 14 3 (1988) 218-240
25. von zur Gathen J., and Gerhard J. Modern Computer Algebra (1999), Cambridge University Press, New York, NY, USA
26. Giusti M., Hägele K., Heintz J., Morais J.-E., Montaña J.-L., and Pardo L.-M. Lower bounds for diophantine approximation. J. Pure Appl. Algebra 117-118 (1997) 277-317
27. Giusti M., Heintz J., Morais J.-E., Morgenstern J., and Pardo L.-M. Straight-line programs in geometric elimination theory. J. Pure Appl. Algebra 124 1-3 (1998) 101-146
28. Giusti M., Heintz J., Morais J.-E., and Pardo L.-M. When polynomial equation systems can be solved fast?. AAECC'11. Lecture Notes in Computer Science vol. 948 (1995), Springer 205-231
29. Giusti M., Lecerf G., and Salvy B. A Gröbner free alternative for polynomial system solving. J. Complexity 17 2 (2001) 154-211
30. Heintz J. Definability and fast quantifier elimination in algebraically closed fields. Theoret. Comput. Sci. 24 3 (1983) 239-277
31. Heintz J., Krick T., Puddu S., Sabia J., and Waissbein A. Deformation techniques for efficient polynomial equation solving. J. Complexity 16 1 (2000) 70-109
32. J. van der Hoeven, Notes on the truncated Fourier transform. Technical Report 2005-5, Université Paris-Sud, Orsay, France, 2005
33. Hubert É. Notes on triangular sets and triangulation-decomposition algorithms. Symbolic and Numerical Scientific Computations. Lecture Notes in Computer Science vol. 2630 (2003), Springer 1-39
34. Jeronimo G., Krick T., Sabia S., and Sombra M. The computational complexity of the Chow form. Found. Comput. Math. 4 1 (2004) 41-117
35. Kalkbrener M. A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. J. Symbolic Comput. 15 2 (1993) 143-167
36. Kalkbrener M. On the complexity of Gröbner bases conversion. J. Symbolic Comput. 28 1-2 (1999) 265-273
37. Lazard D. A new method for solving algebraic systems of positive dimension. Disc. Appl. Math. 33 1-3 (1991) 147-160
38. Lazard D. Solving zero-dimensional algebraic systems. J. Symbolic Comput. 13 2 (1992) 147-160
39. Lecerf G. Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers. J. Complexity 19 4 (2003) 564-596
40. Lecerf G., and Schost É. Fast multivariate power series multiplication in characteristic zero. SADIO Electron. J. Inform. Oper. Res. 5 1 (2003) 1-10
41. F. Lemaire, Contribution à l'algorithmique en algèbre différentielle. Ph.D. thesis, Université Lille I, LIFL, 2002
42. F. Lemaire, M. Moreno Maza, Y. Xie, The RegularChains library, in: Maple Conference, 2005, pp. 355-368
43. Li X., Moreno Maza M., and Schost É. Fast arithmetic for triangular sets: From theory to practice. ISSAC'07 (2007), ACM Press 269-276
44. M. Moreno Maza, Calculs de Pgcd au-dessus des tours d'extensions simples et résolution des systèmes d'équations algébriques. Ph.D. thesis, Université Paris VI, 1997
45. M. Moreno Maza, On triangular decompositions of algebraic varieties. Technical Report TR 4/99, NAG Ltd, Oxford, UK, 1999. in: MEGA'00 Conference, Bath., http://www.csd.uwo.ca/~moreno
46. ∞. J. Symbolic Comput. 28 4-5 (1999) 631-656
47. Pascal C., and Schost É. Change of order for bivariate triangular sets. ISSAC'06 (2006), ACM Press 277-284
48. Recski A. Matroid theory and its applications in electric network theory and in statics (1989), Springer-Verlag, New-York
49. Ritt J.F. Differential Algebra (1966), Dover Publications
50. Schönhage A., and Strassen V. Schnelle Multiplikation großer Zahlen. Computing 7 (1971) 281-292
51. Schost É. Complexity results for triangular sets. J. Symbolic Comput. 36 3-4 (2003) 555-594
52. Schost É. Computing parametric geometric resolutions. Appl. Algebra Eng. Comm. Comput. 13 5 (2003) 349-393
53. Schwartz J.T. Fast probabilistic algorithms for verification of polynomial identities. J. Assoc. Comput. Mach. 27 4 (1980) 701-717
54. Shannon D., and Sweedler M. Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms determine birational equivalence. J. Symbolic Comput. 6 2-3 (1988) 267-273
55. Sommese A.J., Verschelde J., and Wampler C.W. Numerical irreducible decomposition using projections from points on the components. Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering. Contemp. Math. vol. 286 (2001), Amer. Math. Soc. 37-51
56. Tran Q.-N. Efficient Gröbner walk conversion for implicitization of geometric objects. Comput. Aided Geom. Design 21 9 (2004) 837-857
57. Welsh D. Matroid Theory (1976), Academic Press, London
58. Wu W.T. On zeros of algebraic equations - An application of Ritt principle. Kexue Tongbao 31 (1986) 1-5
59. Yang L., and Zhang J. Searching dependency between algebraic equations: An algorithm applied to automated reasoning. Artificial Intelligence in Mathematics (1994), Oxford University Press 147-156
60. Zippel R. Probabilistic algorithms for sparse polynomials. EUROSAM' 79. Lecture Notes in Computer Science vol. 72 (1979), Springer 216-226