## Duke Mathematical Journal

### On the intractability of Hilbert’s Nullstellensatz and an algebraic version of “$NP\not=P$?”

#### Article information

Source
Duke Math. J. Volume 81, Number 1 (1995), 47-54.

Dates
First available: 19 February 2004

http://projecteuclid.org/euclid.dmj/1077245460

Mathematical Reviews number (MathSciNet)
MR1381969

Zentralblatt MATH identifier
0882.03040

Digital Object Identifier
doi:10.1215/S0012-7094-95-08105-8

#### Citation

Shub, Michael; Smale, Steve. On the intractability of Hilbert’s Nullstellensatz and an algebraic version of “ N P ≠ P ? ”. Duke Mathematical Journal 81 (1995), no. 1, 47--54. doi:10.1215/S0012-7094-95-08105-8. http://projecteuclid.org/euclid.dmj/1077245460.

#### References

• [1] L. Blum, M. Shub, and S. Smale, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines, Bull. Amer. Math. Soc. 21 (1989), no. 1, 1–46.
• [2] W. D. Brownawell, Bounds for the degrees in the Nullstellensatz, Ann. of Math. (2) 126 (1987), no. 3, 577–591.
• [3] F. Cucker, M. Shub, and S. Smale, Separation of complexity classes in Koiran's weak model, to appear in Theoret. Comput. Sci.
• [4] J. Heintz, On the computational complexity of polynomials and bilinear mappings: A survey, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (Menorca, 1987), Lecture Notes in Comput. Sci., vol. 356, Springer-Verlag, Berlin, 1989, pp. 269–300.
• [5] J. Heintz and J. Morgenstern, On the intrinsic complexity of elimination theory, J. Complexity 9 (1993), no. 4, 471–498.
• [6] W. de Melo and B. F. Svaiter, The cost of computing integers, to appear in Proc. Amer. Math. Soc.
• [7] C. G. Moreira, On asymptotic estimates for arithmetic cost functions, to appear.
• [8] M. Shub, Some remarks on Bezout's Theorem and complexity theory, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990) eds. M. W. Hirsch, J. E. Marsden, and M. Shub, Springer-Verlag, New York, 1993, pp. 443–455.
• [9] L. G. Valiant, Completeness classes in algebra, Conference Record of the Eleventh Annual ACM Symposium on Theory of Computing (Atlanta, Ga., 1979), ACM, New York, 1979, pp. 249–261.