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[115] A Remark on Integral Geometry

  • Raoul Bott
  • Cliff Taubes
Chapter
Part of the Contemporary Mathematicians book series (CM)

Introduction

The occasion for this note is the following result of Shub and Smale [SS] on the average number of solutions of polynomial equations over the reals.

Suppose that {f1,…,fn} are nhomogeneous polynomials in(n+1)variables(x0…,xn).Then generically the simultaneous solutions of the equations

$${f_i}\left( x \right)\, = \,0\,,\,\,\,\,\,\,\,\,\,i\, = \,1, \ldots ,n$$

References

  1. ATIYAH, M. F., R. BOTI and V. K. PATODI. On the heat equation and theindex theorem. Invent. Math. 19 (1973), 279- 330.MathSciNetCrossRefGoogle Scholar
  2. EDELMAN, A. and E. KOSTLAN. How many zeros of a random polynomialare real?Bull. Amer. Math. Soc. N.S. 32(1) (1995), 1-37.MathSciNetCrossRefGoogle Scholar
  3. SANTALÓ, L.A. Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and its applications Vol. 1, Addison Wesley, 1976.Google Scholar
  4. SMALE, S. and M. SHUB. Complexity of Bezout’s theorem. II. Volumes and probabilities. Computational Algebraic Geometry (Nice, 1992), Progr. Math. 109, Birkhäuser (1993), 267-285MathSciNetzbMATHGoogle Scholar
  5. WEYL, H. The Classical Groups.Princeton Univ. Press, 1946.zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Raoul Bott
    • 1
  • Cliff Taubes
    • 1
  1. 1.Harvard University, Dept. of MathematicsCambridgeUSA

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