Abstract
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.
Keywords
- Condition Number
- Projective Space
- Unit Sphere
- Unitary Group
- Frobenius Norm
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.
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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DOI: https://doi.org/10.1007/978-1-4612-2752-6_19
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7652-4
Online ISBN: 978-1-4612-2752-6
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