Complexity of Bezout’s Theorem II Volumes and Probabilities

  • Michael Shub
  • Steve Smale
Part of the Progress in Mathematics book series (PM, volume 109)


In this paper we study volume estimates in the space of systems of n homegeneous polynomial equations of fixed degrees d i with respect to a natural Hermitian structure on the space of such systems invariant under the action of the unitary group. We show that the average number of real roots of real systems is D 1/2 where D = Π d i is the Be zout number. We estimate the volume of the subspace of badly conditioned problems and show that volume is bounded by a small degree polynomial in n, N and D times the reciprocal of the condition number to the fourth power. Here N is the dimension of the space of systems.


Condition Number Projective Space Unit Sphere Unitary Group Frobenius Norm 
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Copyright information

© Birkhäuser Boston 1993

Authors and Affiliations

  • Michael Shub
    • 2
  • Steve Smale
    • 1
  1. 1.T.J. Watson Research CenterIBMYorktown HeightsUSA
  2. 2.Math DepartmentUniversity of CaliforniaBerkeleyUSA

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