Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



The fundamental theorem of algebra and complexity theory

Author: Steve Smale
Journal: Bull. Amer. Math. Soc. 4 (1981), 1-36
MSC (1980): Primary 00-02, 12D10, 68C25, 65H05, 58-02; Secondary 01A05, 30D10
MathSciNet review: 590817
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • B. Barna, 1956, Über die Divergenzpunkte des Newtonsches Verfahrens zur Bestimmung von Wurzeln Algebraischen Gleichungen. II, Publicatione Mathematicae, Debrecen, vol. 4, pp. 384-397. MR 78956
  • L. E. J. Brouwer, 1924 (w. B. de Loor), Intuitionischer Beweis des Fundamentalsatzes der Algebra, Coll. Works, vol. 1 (1975), North-Holland, Amsterdam.
  • G. Collins, 1977, Infallible calculation of polynomials to specified precision in mathematical software. III, Academic Press, New York. MR 474768
  • G. Debreu, 1959, Theory of value, Yale Univ. Press, New Haven, Conn.
  • B. Dejon and P. Henrici, 1969, Constructive aspects of the fundamental theorem of algebra, Wiley, New York. MR 253555
  • C. Eaves and H. Scarf, 1976, The solution of systems of piecewise linear equations, Math. Operations Res., vol. 1, pp. 1-27. MR 445792
  • H. Eves, 1976, An introduction to the history of mathematics (4th ed.), Holt, Rinehart and Winston, New York.
  • Michael R. Garey and David S. Johnson, Computers and intractability, W. H. Freeman and Co., San Francisco, Calif., 1979. A guide to the theory of NP-completeness; A Series of Books in the Mathematical Sciences. MR 519066
  • C. F. Gauss, 1973, Werke, Band X, Georg Olms Verlag, New York.
  • Phillip A. Griffiths, Complex differential and integral geometry and curvature integrals associated to singularities of complex analytic varieties, Duke Math. J. 45 (1978), no. 3, 427–512. MR 507455
  • J. Hartmanis, Observations about the development of theoretical computer science, 20th Annual Symposium on Foundations of Computer Science (San Juan, Puerto Rico, 1979) IEEE, New York, 1979, pp. 224–233. MR 598111
  • W. Hayman, 1958, Multivalent functions, Cambridge Univ. Press, Cambridge, England. MR 108586
  • P. Henrici, 1977, Applied and computational complex analysis, Wiley, New York. MR 453984
  • E. Hille, 1962, Analytic function theory. II, Ginn, Boston. MR 201608
  • M. Hirsch, 1963, A proof of the non-retractability of a cell onto its boundary, Proc. Amer. Math. Soc. 14, pp. 364-365. MR 145502
  • Morris W. Hirsch and Stephen Smale, On algorithms for solving 𝑓(𝑥)=0, Comm. Pure Appl. Math. 32 (1979), no. 3, 281–313. MR 517937,
  • David A. Hoffman and Robert Osserman, The geometry of the generalized Gauss map, Mem. Amer. Math. Soc. 28 (1980), no. 236, iii+105. MR 587748,
  • Harold Hotelling, Tubes and Spheres in n-Spaces, and a Class of Statistical Problems, Amer. J. Math. 61 (1939), no. 2, 440–460. MR 1507387,
  • W. Hurewicz, 1958, Lectures on ordinary differential equations, MIT Press, Cambridge, Mass. MR 103305
  • J. Jenkins, 1965, Univalent functions and conformal mapping, Springer, New York. MR 96806
  • R. Kellog, T. Li and J. Yorke, 1976, A constructive proof of the Brouwer fixed point theorem and computational results, SIAM J. Numer. Anal. 13, pp. 473-483. MR 416010
  • I. Lakatos, 1976, Proofs and refutations, Cambridge Univ. Press, Cambridge, England. MR 479916
  • S. Lang, 1965, Algebra, Addison-Wesley, Reading, Mass. MR 197234
  • M. Marden, 1966, Geometry of polynomials, Math. Surveys, no. 3, Amer Math. Soc., Providence, R. I. MR 225972
  • A. Ostrowski, 1973, Solutions of equations in Euclidean and Banach spaces, Academic Press, New York. MR 359306
  • Jean-Claude Pont, 1974, La topologie algébrique, des origine à Poincaré, Presses Universitaires de France, Paris. MR 472374
  • L. Santalo, 1976, Integral geometry and geometric probability, Addison-Wesley, Reading, Mass. MR 433364
  • H. Scarf, 1973, The computation of economic equilibria (in collaboration with T. Hansen) Yale Univ. Press, New Haven, Conn. MR 391909
  • Glenn Schober, Coefficient estimates for inverses of schlicht functions, Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979) Academic Press, London-New York, 1980, pp. 503–513. MR 623495
  • S. Smale, 1974, Sufficient conditions for an optimum, Dynamical Systems-Warwick 1974, Lecture Notes in Math., vol. 468, Springer-Verlag, Berlin and New York. MR 649781
  • S. Smale, 1976, A convergent process of price adjustment and global Newton methods, J. Math. Econom., 3, pp. 107-120. MR 411577
  • S. Smale, (to appear), Global analysis and economics, Handbook of Mathematical Economics (Arrow and Intrilligator, eds.), North-Holland, Amsterdam. MR 459553
  • D. Smith, 1953, History of mathematics, vol. II, Dover, New York.
  • D. Struik, 1969, A source book in mathematics, 1200-800, Harvard Univ. Press, Cambridge, Mass. MR 238647
  • J. Traub, ed., 1976, Analytic computational complexity, Academic Press, New York. MR 395140
  • B. Van der Waerden, 1953, Modern algebra, Vol. I, Ungar, New York.
  • B. Van der Waerden, 1950, Modern algebra, Vol. II, Ungar, New York. MR 29363
  • Y.-H. Wan, 1975, On local Pareto optima, J. Math. Econom., 2, pp. 35-42. MR 401106
  • H. Weyl, Randbemerkungen zu Hauptproblemen der Mathematik, Math. Z. 20 (1924), no. 1, 131–150 (German). MR 1544667,
  • Hermann Weyl, On the Volume of Tubes, Amer. J. Math. 61 (1939), no. 2, 461–472. MR 1507388,
  • H. Whitney, 1972, Complex analytic varieties, Addison-Wesley, Reading, Mass. MR 387634
  • J. H. Wilkinson, 1963, Rounding errors in algebraic processes, Prentice-Hall, Englewood Cliffs, N.J. MR 161456
  • Philip Wolfe, The “ellipsoid algorithm”, Science 208 (1980), no. 4441, 240–242. MR 565382,

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1980): 00-02, 12D10, 68C25, 65H05, 58-02, 01A05, 30D10

Retrieve articles in all journals with MSC (1980): 00-02, 12D10, 68C25, 65H05, 58-02, 01A05, 30D10

Additional Information


American Mathematical Society