Foundations of Computational Mathematics

, Volume 19, Issue 1, pp 131–157 | Cite as

Probabilistic Condition Number Estimates for Real Polynomial Systems I: A Broader Family of Distributions

  • Alperen A. ErgürEmail author
  • Grigoris Paouris
  • J. Maurice Rojas


We consider the sensitivity of real roots of polynomial systems with respect to perturbations of the coefficients. In particular—for a version of the condition number defined by Cucker and used later by Cucker, Krick, Malajovich, and Wschebor—we establish new probabilistic estimates that allow a much broader family of measures than considered earlier. We also generalize further by allowing overdetermined systems. In Part II, we study smoothed complexity and how sparsity (in the sense of restricting which terms can appear) can help further improve earlier condition number estimates.


Condition number Epsilon net Probabilistic bound Kappa Real-solving Overdetermined Subgaussian 

Mathematics Subject Classification

Primary 65Y20 Secondary 51F99 68Q25 



The Authors would like to thank the anonymous referees for detailed remarks that greatly helped clarify our paper.


  1. 1.
    Franck Barthe and Alexander Koldobsky, “Extremal slabs in the cube and the Laplace transform,” Adv. Math. 174 (2003), pp. 89–114.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carlos Beltrán and Luis-Miguel Pardo, “Smale’s 17th problem: Average polynomial time to compute affine and projective solutions,” Journal of the American Mathematical Society 22 (2009), pp. 363–385.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lenore Blum, Felipe Cucker, Mike Shub, and Steve Smale, Complexity and Real Computation, Springer-Verlag, 1998.Google Scholar
  4. 4.
    Jean Bourgain, “On the isotropy-constant problem for \(\psi _2\) -bodies,” in Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics, vol. 1807, pp. 114–121, Springer Berlin Heidielberg, 2003.Google Scholar
  5. 5.
    Peter Bürgisser and Felipe Cucker, “On a problem posed by Steve Smale,” Annals of Mathematics, pp. 1785–1836, Vol. 174 (2011), no. 3.Google Scholar
  6. 6.
    Peter Bürgisser and Felipe Cucker, Condition, Grundlehren der mathematischen Wissenschaften, no. 349, Springer-Verlag, 2013.Google Scholar
  7. 7.
    John F. Canny, The Complexity of Robot Motion Planning, ACM Doctoral Dissertation Award Series, MIT Press, 1987.Google Scholar
  8. 8.
    Peter Bürgisser; Felipe Cucker; and Pierre Lairez, “Computing the homology of basic semialgebraic sets in weakly exponential time,” Math ArXiV preprint arXiv:1706.07473.
  9. 9.
    D. Castro, Juan San Martín, Luis M. Pardo, “Systems of Rational Polynomial Equations have Polynomial Size Approximate Zeros on the Average,” Journal of Complexity 19 (2003), pp. 161–209.Google Scholar
  10. 10.
    Felipe Cucker, “Approximate zeros and condition numbers,” Journal of Complexity 15 (1999), no. 2, pp. 214–226.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Felipe Cucker, Teresa Krick, Gregorio Malajovich, and Mario Wschebor, “A numerical algorithm for zero counting I. Complexity and accuracy,” J. Complexity 24 (2008), no. 5–6, pp. 582–605MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Felipe Cucker, Teresa Krick, Gregorio Malajovich, and Mario Wschebor, “A numerical algorithm for zero counting II. Distance to ill-posedness and smoothed analysis,” J. Fixed Point Theory Appl. 6 (2009), no. 2, pp. 285–294.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Felipe Cucker, Teresa Krick, Gregorio Malajovich, and Mario Wschebor, “A numerical algorithm for zero counting III: Randomization and condition,” Adv. in Appl. Math. 48 (2012), no. 1, pp. 215–248.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Felipe Cucker; Teresa Krick; and Mike Shub, “Computing the Homology of Real Projective Sets,” Found. Comp. Math., to appear. (Earlier version available as Math ArXiV preprint arXiv:1602.02094.)
  15. 15.
    Nikos Dafnis and Grigoris Paouris, “Small ball probability estimates, \(\Psi _2\) -behavior and the hyperplane conjecture,” Journal of Functional Analysis 258 (2010), pp. 1933–1964.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jean-Piere Dedieu, Mike Shub, “Newton’s Method for Overdetermined Systems Of Equations,” Math. Comp. 69 (2000), no. 231, pp. 1099–1115.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    James Demmel, Benjamin Diament, and Gregorio Malajovich, “On the Complexity of Computing Error Bounds,” Found. Comput. Math. pp. 101–125 (2001).Google Scholar
  18. 18.
    Wassily Hoeffding, “Probability inequalities for sums of bounded random variables,” Journal of the American Statistical Association, 58 (301):13–30, 1963.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    O. D. Kellog, “On bounded polynomials in several variables,” Mathematische Zeitschrift, December 1928, Volume 27, Issue 1, pp. 55–64.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Bo’az Klartag and Emanuel Milman, “Centroid bodies and the logarithmic Laplace Transform – a unified approach,” J. Func. Anal., 262(1):10–34, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Alexander Koldobsky and Alain Pajor, “A Remark on Measures of Sections of \(L_p\) -balls,” Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics 2169, pp. 213–220, Springer-Verlag, 2017.Google Scholar
  22. 22.
    Eric Kostlan, “On the Distribution of Roots of Random Polynomials,” Ch. 38 (pp. 419–431) of From Topology to Computation: Proceedings of Smalefest (M. W. Hirsch, J. E. Marsden, and M. Shub, eds.), Springer-Verlag, New York, 1993.Google Scholar
  23. 23.
    Pierre Lairez, “A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time,” Foundations of Computational Mathematics,
  24. 24.
    G. Livshyts, Grigoris Paouris and P. Pivovarov, “Sharp bounds for marginal densities of product measures,” Israel Journal of Mathematics, Vol. 216, Issue 2, pp. 877–889.Google Scholar
  25. 25.
    G. Paouris and P. Pivovarov, “Randomized Isoperimetric Inequalities”, IMA Volume “Discrete Structures: Analysis and Applications” (Springer)Google Scholar
  26. 26.
    Hoi H. Nguyen, “On a condition number of general random polynomial systems,” Mathematics of Computation (2016) 85, pp. 737–757MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mark Rudelson and Roman Vershynin, “The Littlewood-Offord Problem and Invertibility of Random Matrices,” Adv. Math. 218 (2008), no. 2, pp. 600–633.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mark Rudelson and Roman Vershynin, “The Smallest Singular Value of Rectangular Matrix,” Communications on Pure and Applied Mathematics 62 (2009), pp. 1707–1739.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mark Rudelson and Roman Vershynin, “Small ball Probabilities for Linear Images of High-Dimensional Distributions,” Int. Math. Res. Not. (2015), no. 19, pp. 9594–9617.Google Scholar
  30. 30.
    Roman Vershynin , “High Dimensional Probability: An Introduction with Application in Data Science”, available at
  31. 31.
    Roman Vershynin , “Four Lectures on Probabilistic Methods for Data Science”, available at
  32. 32.
    Igor R. Shafarevich, Basic Algebraic Geometry 1: Varieties in Projective Space, 3rd edition, Springer-Verlag (2013).Google Scholar
  33. 33.
    Mike Shub and Steve Smale, “Complexity of Bezout’s Theorem I. Geometric Aspects,” J. Amer. Math. Soc. 6 (1993), no. 2, pp. 459–501.MathSciNetzbMATHGoogle Scholar
  34. 34.
    Roman Vershynin, “Introduction to the Non-Asymptotic Analysis of Random Matrices,” Compressed sensing, pp. 210–268, Cambridge Univ. Press, Cambridge, 2012.Google Scholar
  35. 35.
    Assaf Naor and Artem Zvavitch, “Isomorphic embedding of \(\ell _p^n\), \(1 < p < 2\), into \(\ell _1^{(1+\varepsilon )n}\) ,” Israel J. Math. 122 (2001), pp. 371–380.Google Scholar

Copyright information

© SFoCM 2018

Authors and Affiliations

  • Alperen A. Ergür
    • 1
    Email author
  • Grigoris Paouris
    • 2
  • J. Maurice Rojas
    • 2
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Department of MathematicsTexas A&M University TAMU 3368College StationUSA

Personalised recommendations