Bernoulli

  • Bernoulli
  • Volume 15, Number 1 (2009), 249-266.

Random systems of polynomial equations. The expected number of roots under smooth analysis

Diego Armentano and Mario Wschebor

Full-text: Open access

Abstract

We consider random systems of equations over the reals, with $m$ equations and $m$ unknowns $P_i(t)+X_i(t)=0, t∈ℝ^m, i=1, …, m$, where the $P_i$’s are non-random polynomials having degrees $d_i$’s (the “signal”) and the $X_i$’s (the “noise”) are independent real-valued Gaussian centered random polynomial fields defined on $ℝ^m$, with a probability law satisfying some invariance properties.

For each $i, P_i$ and $X_i$ have degree $d_i$.

The problem is the behavior of the number of roots for large $m$. We prove that under specified conditions on the relation signal over noise, which imply that in a certain sense this relation is neither too large nor too small, it follows that the quotient between the expected value of the number of roots of the perturbed system and the expected value corresponding to the centered system (i.e., $P_i$ identically zero for all $i=1, …, m$), tends to zero geometrically fast as $m$ tends to infinity. In particular, this means that the behavior of this expected value is governed by the noise part.

Article information

Source
Bernoulli, Volume 15, Number 1 (2009), 249-266.

Dates
First available in Project Euclid: 3 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.bj/1233669890

Digital Object Identifier
doi:10.3150/08-BEJ149

Mathematical Reviews number (MathSciNet)
MR2546806

Zentralblatt MATH identifier
1203.60057

Keywords
random polynomials Rice formula system of random equations

Citation

Armentano, Diego; Wschebor, Mario. Random systems of polynomial equations. The expected number of roots under smooth analysis. Bernoulli 15 (2009), no. 1, 249--266. doi:10.3150/08-BEJ149. https://projecteuclid.org/euclid.bj/1233669890


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References

  • [1] Azaïs, J.-M. and Wschebor, M. (2005). On the roots of a random system of equations. The theorem of Shub and Smale and some extensions., Found. Comput. Math. 5 125–144.
  • [2] Azaïs, J.-M. and Wschebor, M. (2009). Level sets and extrema of random processes and fields. John Wiley and Sons. To, appear.
  • [3] Bharucha-Reid, A.T. and Sambandham, M. (1986)., Random Polynomials. Orlando, FL: Academic Press.
  • [4] Dedieu, J.-P. and Malajovich, G. (2008). On the number of minima of a random polynomial., J. Complexity 24 89–108.
  • [5] Edelman, A. and Kostlan, E. (1995). How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. (N.S.) 32 1–37.
  • [6] Kac, M. (1943). On the average number of real roots of a random algebraic equation., Bull. Amer. Math. Soc. 49 314–320.
  • [7] Malajovich, G. and Maurice Rojas, J. (2004). High probability analysis of the condition number of sparse polynomial systems., Theoret. Comput. Sci. 315 524–555.
  • [8] McLennan, A. (2002). The expected number of real roots of a multihomogeneous system of polynomial equations., Amer. J. Math. 124 49–73.
  • [9] Maurice Rojas, J. (1996). On the average number of real roots of certain random sparse polynomial systems. In, The Mathematics of Numerical Analysis (Park City, UT, 1995) 32. Lectures in Appl. Math. 689–699. Providence, RI: Amer. Math. Soc.
  • [10] Shub, M. and Smale, S. (1993). Complexity of Bezout’s theorem, II. Volumes and probabilities. In, Computational Algebraic Geometry (Nice, 1992). Progr. Math. 109 267–285. Boston, MA: Birkhäuser Boston.
  • [11] Wschebor, M. (2005). On the Kostlan–Shub–Smale model for random polynomial systems. Variance of the number of roots., J. Complexity 21 773–789.