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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Some of this work was carried out when Shub was visiting the Berkeley Math Department for 2 months in 1992,
Supported partially by NSF funds
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© 1993 Birkhäuser Boston
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Shub, M., Smale, S. (1993). Complexity of Bezout’s Theorem II Volumes and Probabilities. In: Eyssette, F., Galligo, A. (eds) Computational Algebraic Geometry. Progress in Mathematics, vol 109. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2752-6_19
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Print ISBN: 978-1-4612-7652-4
Online ISBN: 978-1-4612-2752-6
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