# On Smale’s Work in the Theory of Computation: From Polynomial Zeros to Continuous Complexity

Chapter

## Abstract

I am delighted to have this opportunity to comment on the work of Steve Smale. Mike Shub was kind enough to give me an advance copy of his excellent paper (Shub [93]), which was very useful for it served as a reminder of the many areas in the theory of computation to which Steve has made major or often seminal contributions.

## Keywords

Computational Complexity Turing Machine Digital Computer Intrinsic Difficulty Multivariate Integration
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [89]BLUM, L, SHUB, M., AND SMALE, S. On a Theory of Computation and Complexity Over the Real Numbers:
*NP*-completeness, Recursive Functions and Universal Machines,*Bull Amer. Math. Soc*. 21 (1989), 1–46MathSciNetMATHCrossRefGoogle Scholar - [89]RENEGAR, J. On the Worst Case Arithmetic Complexity of Approximating Zeros of Systems of Polynomials,
*SIAM J. Comput*. 18 (1989), 350–370.MathSciNetMATHCrossRefGoogle Scholar - [93]SHUB, M. On the Work of Steve Smale on the Theory of Computation. This volume.Google Scholar
- [85]SHUB, M. AND SMALE, S. Computational Complexity, On the Geometry of Polynomials and a Theory of Cost: Part I,
*Ann. Scient. Ecole. Norm. Sup*. 18 (1985), 107–142.MathSciNetMATHGoogle Scholar - [86]SHUB, M. AND SMALE, S. Computational Complexity, On the Geometry of Polynomials and a Theory of Cost: Part II,
*SIAM J. Comput*. 15 (1986), 145–161.MathSciNetMATHCrossRefGoogle Scholar - [81]SMALE, S. The Fundamental Theorem of Algebra and Complexity Theory,
*Bull. Amer. Math. Soc*. 4 (1981), 1–36.MathSciNetMATHCrossRefGoogle Scholar - [83]SMALE, S.On the Average Number of Steps in the Simplex Method of Linear Programming,
*Math. Programming*27 (1983), 242–262.MathSciNetCrossRefGoogle Scholar - [88]TRAUB, J.F., WASILKOWSKI, G.W., AND WOUźNIAKOWSKI, H.
*Information-Based Complexity*, Academic Press, Boston, MA, 1988.MATHGoogle Scholar - [91]TRAUB, J.F. AND WOUźNIAKOWSKI, H. Information-Based Complexity: New Questions for Mathematicians,
*Math. Intelligencer*13 (1991), 34–43.MathSciNetMATHCrossRefGoogle Scholar - [91]WERSCHULZ, A.G.Computational Complexity of Differential and Integral Equations, Oxford University Press, Oxford, 1991.MATHGoogle Scholar

## Copyright information

© Springer-Verlag New York, Inc. 1993