Abstract
In the forthcoming paper of Beltrán and Pardo, the average complexity of linear homotopy methods to solve polynomial equations with random initial input (in a sense to be described below) was proven to be finite, and even polynomial in the size of the input. In this paper, we prove that some other higher moments are also finite. In particular, we show that the variance is polynomial in the size of the input.
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C. Beltrán, L.M. Pardo, On Smale’s 17th problem: a probabilistic positive solution, Found. Comput. Math. 8(1), 1–43 (2008).
C. Beltrán, L.M. Pardo, Smale’s 17th problem: average polynomial time to compute affine and projective solutions, J. Am. Math. Soc. 22, 363–385 (2009).
C. Beltrán, L.M. Pardo, Fast linear homotopy to find approximate zeros of polynomial systems (to appear).
C. Beltrán, L.M. Pardo, Computing several zeros of polynomial systems: a complexity analysis and Shannon’s Entropy (to appear).
L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation (Springer, New York, 1998).
D. Coppersmith, C.A. Neff, Roots of a polynomial and its derivatives, in Proceedings of the Fifth Annual ACM–SIAM Symposium on Discrete Algorithms (Arlington, VA, 1994) (ACM, New York, 1994), pp. 271–279.
M.H. Kim, S. Sutherland, Polynomial root-finding algorithms and branched covers, SIAM J. Comput. 23(2), 415–436 (1994).
M.H. Kim, M. Martens, S. Sutherland, A universal bound for the average cost of rootfinding. Preprint.
V.Y. Pan, Solving a polynomial equation: some history and recent progress, SIAM Rev. 39(2), 187–220 (1997).
J. Renegar, On the worst-case arithmetic complexity of approximating zeros of polynomials, J. Complex. 3(2), 90–113 (1987).
W. Rudin, Real and Complex Analysis, 3rd edn. (McGraw-Hill Book, New York, 1987).
L.A. Santaló, Integral Geometry and Geometric Probability (Addison-Wesley, Reading, 1976).
M. Shub, Some remarks on Bezout’s theorem and complexity theory, in From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), ed. by M.W. Hirsch, J.E. Marsden, M. Shub (Springer, New York, 1993), pp. 443–455.
M. Shub, Complexity of Bézout’s theorem. VI: Geodesics in the condition (number) metric, Found. Comput. Math. 9(2), 171–178 (2009).
M. Shub, S. Smale, Complexity of Bézout’s theorem. I. Geometric aspects, J. Am. Math. Soc. 6(2), 459–501 (1993).
M. Shub, S. Smale, Complexity of Bezout’s theorem. V. Polynomial time, Theor. Comput. Sci. 133(1), 141–164 (1994). Selected papers of the Workshop on Continuous Algorithms and Complexity (Barcelona, 1993).
S. Smale, Complexity theory and numerical analysis, in Acta Numerica, 1997. Acta Numer., vol. 6 (Cambridge University Press, Cambridge, 1997), pp. 523–551.
S. Smale, Mathematical problems for the next century, in Mathematics: Frontiers and Perspectives (American Mathematical Society, Providence, 2000), pp. 271–294.
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Communicated by Michael Todd.
C. Beltrán was partially supported by MTM2007-62799 and a Spanish postdoctoral grant (FECYT).
M. Shub was partially supported by an NSERC Discovery Grant.
Thanks to one of the referees for his many comments that helped to improve the first version of this manuscript.
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Beltrán, C., Shub, M. A Note on the Finite Variance of the Averaging Function for Polynomial System Solving. Found Comput Math 10, 115–125 (2010). https://doi.org/10.1007/s10208-009-9054-4
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DOI: https://doi.org/10.1007/s10208-009-9054-4