The multivariate resultant is NP-hard in any characteristic

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 6281 LNCS, Issue , 2010, Pages 477-488

  Grenet, Bruno ^a,b   Koiran, Pascal ^a,b   Portier, Natacha ^a,b  

^a UNIV LYON   (France)

^b UNIVERSITY OF TORONTO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

ARTIFICIAL INTELLIGENCE; COMPUTER SCIENCE; COMPUTERS;

COMPUTATIONAL ALGEBRAIC GEOMETRY; FUNDAMENTAL TOOLS; GROUND FIELD; HOMOGENEOUS EQUATIONS; LOW DEGREE; NP-HARDNESS RESULTS; RIEMANN HYPOTHESIS; UPPER BOUND;

COMPUTATIONAL GEOMETRY;

EID: 78349258141     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-642-15155-2_42     Document Type: Conference Paper

References (34)

1. Arora, S., Barak, B.: Computational Complexity: A Modern Approach, 1st edn. Cambridge University Press, Cambridge (2009)
2. Busé, L., D'Andrea, C.: On the irreducibility of multivariate subresultants. CR Math. 338(4), 287-290 (2004)
3. Canny, J.F.: A new algebraic method for robot motion planning and real geometry. In: Proc. FOCS 1987, pp. 39-48 (1987)
4. Canny, J.F.: The complexity of robot motion planning. In: ACM Doctoral Dissertation Award, vol. 1987. MIT Press, Cambridge (1988)
5. Canny, J.F.: Some algebraic and geometric computations in PSPACE. In: Proc. STOC 1988, pp. 460-469 (1988)
6. Canny, J.F.: Generalized characteristic polynomials. In: Gianni, P. (ed.) ISSAC 1988. LNCS, vol. 358. Springer, Heidelberg (1989)
7. Canny, J.F., Kaltofen, E., Yagati, L.: Solving systems of nonlinear polynomial equations faster. In: Proc. SIGSAM 1989, pp. 121-128 (1989)
8. Canny, J.F., Reif, J.H.: New lower bound techniques for robot motion planning problems. In: Proc. FOCS 1987, pp. 49-60 (1987)
9. Cattani, E., Dickenstein, A.: Introduction to residues and resultants. In: Dickenstein, A., Emiris, I. (eds.) Solving Polynomial Equations, pp. 1-61. Springer, Heidelberg (2005)
10. D'Andrea, C., Dickenstein, A.: Explicit formulas for the multivariate resultant. J. Pure Appl. Algebra 164(1-2), 59-86 (2001)
11. Dixon, A.: The eliminant of three quantics in two independent variables. Proc. Lond. Math. Soc. 6, 468-478 (1908)
12. Emiris, I., Mourrain, B.: Matrices in elimination theory. J. Symb. Comput. 28(1-2), 3-43 (1999)
13. Galperin, H., Wigderson, A.: Succinct representations of graphs. Inform. Control 56(3), 183-198 (1984) (Pubitemid 14564551)
14. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Series of Books in the Mathematical Sciences. W.H. Freeman, New York (1979)
15. Grenet, B.: Difficulté du résultant et des grands déterminants. Technical report, RRLIP2009-32, LIP (2009), http://prunel.ccsd.cnrs.fr/ensl-00431714/
16. Grenet, B., Koiran, P., Portier, N.: The multivariate resultant is NP-hard in any characteristic. Technical report, RRLIP2009-34, LIP (2009), http://prunel.ccsd.cnrs.fr/ensl-00440842/
17. Ierardi, D.: Quantifier elimination in the theory of an algebraically-closed field. In: Proc. STOC 1989, pp. 138-147 (1989)
18. Jeronimo, G., Sabia, J.: Computing multihomogeneous resultants using straight-line programs. J. Symb. Comput. 42(1-2), 218-235 (2007)
19. Kaltofen, E., Koiran, P.: On the complexity of factoring bivariate supersparse (lacunary) polynomials. In: Proc. ISSAC 2005, pp. 208-215. ACM, New York (2005)
20. Kaltofen, E., Koiran, P.: Expressing a fraction of two determinants as a determinant. In: Proc. ISSAC 2008, pp. 141-146 (2008)
21. Kapur, D., Saxena, T.: Comparison of various multivariate resultant formulations. In: Proc. ISSAC 1995, pp. 187-194 (1995)
22. Kapur, D., Saxena, T., Yang, L.: Algebraic and geometric reasoning using Dixon resultants. In: Proc. ISSAC 1994, pp. 99-107 (1994)
23. Koiran, P.: Hilbert's Nullstellensatz is in the polynomial hierarchy. J. Complexity 12(4), 273-286 (1996)
24. Koiran, P.: The complexity of local dimensions for constructible sets. J. Complexity 16(1), 311-323 (2000)
25. Lazard, D.: Résolution des systémes d'èquations algébriques. Theor. Comput. Sci. 15(1), 77-110 (1981)
26. Macaulay, F.S.: Some formulae in elimination. Proc. Lond. Math. Soc. 1(1), 3 (1902)
27. Macaulay, F.S.: The Algebraic Theory of Modular Systems. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1994)
28. Plaisted, D.A.: New NP-hard and NP-complete polynomial and integer divisibility problems. Theor. Comput. Sci. 31(1-2), 125-138 (1984)
29. Renegar, J.: On the worst-case arithmetic complexity of approximating zeros of systems of polynomials. SIAM J. Comput. 18, 350 (1989)
30. Seidenberg, A.: A new decision method for elementary algebra. Ann. Math. 60(2), 365-374 (1954)
31. Shoup, V.: New algorithms for finding irreducible polynomials over finite fields. Math. Comput. 54(189), 435-447 (1990)
32. Sturmfels, B.: Sparse elimination theory. In: Eisenbud, D., Robbiano, L. (eds.) Proc. Comput. Algebr. Geom. Commut. Algebra (1991)
33. van der Waerden, B.L.: Modern Algebra, 3rd edn. F. Ungar Publishing Co, New York (1950)
34. von zur Gathen, J., Karpinski, M., Shparlinski, I.: Counting curves and their projections. Comput. Complex 6(1), 64-99 (1996)