Asymptotic variance of the number of real roots of random polynomial systems

Armentano, D.; Azaïs, J-M.; Dalmao, F.; León, J.

doi:10.1090/proc/14215

Asymptotic variance of the number of real roots of random polynomial systems
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by D. Armentano, J-M. Azaïs, F. Dalmao and J. R. León PDF

Proc. Amer. Math. Soc. 146 (2018), 5437-5449 Request permission

Abstract:

We obtain the asymptotic variance, as the degree goes to infinity, of the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size. Our main tools are the Kac-Rice formula for the second factorial moment of the number of roots and a Hermite expansion of this random variable.

References

Jean-Marc Azaïs and Mario Wschebor, Level sets and extrema of random processes and fields, John Wiley & Sons, Inc., Hoboken, NJ, 2009. MR 2478201, DOI 10.1002/9780470434642
Jean-Marc Azaïs and Mario Wschebor, On the roots of a random system of equations. The theorem on Shub and Smale and some extensions, Found. Comput. Math. 5 (2005), no. 2, 125–144. MR 2149413, DOI 10.1007/s10208-004-0119-0
A. T. Bharucha-Reid and M. Sambandham, Random polynomials, Probability and Mathematical Statistics, Academic Press, Inc., Orlando, FL, 1986. MR 856019
Federico Dalmao, Asymptotic variance and CLT for the number of zeros of Kostlan Shub Smale random polynomials, C. R. Math. Acad. Sci. Paris 353 (2015), no. 12, 1141–1145 (English, with English and French summaries). MR 3427922, DOI 10.1016/j.crma.2015.09.016
Alan Edelman and Eric Kostlan, How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 1, 1–37. MR 1290398, DOI 10.1090/S0273-0979-1995-00571-9
Anne Estrade and José R. León, A central limit theorem for the Euler characteristic of a Gaussian excursion set, Ann. Probab. 44 (2016), no. 6, 3849–3878. MR 3572325, DOI 10.1214/15-AOP1062
Kambiz Farahmand, Topics in random polynomials, Pitman Research Notes in Mathematics Series, vol. 393, Longman, Harlow, 1998. MR 1679392
Herbert Federer, Two theorems in geometric measure theory, Bull. Amer. Math. Soc. 72 (1966), 719. MR 196033, DOI 10.1090/S0002-9904-1966-11567-7
E. Kostlan, On the distribution of roots of random polynomials, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990) Springer, New York, 1993, pp. 419–431. MR 1246137
Marie F. Kratz and José R. León, Hermite polynomial expansion for non-smooth functionals of stationary Gaussian processes: crossings and extremes, Stochastic Process. Appl. 66 (1997), no. 2, 237–252. MR 1440400, DOI 10.1016/S0304-4149(96)00122-6
Antonio Lerario and Erik Lundberg, On the zeros of random harmonic polynomials: the truncated model, J. Math. Anal. Appl. 438 (2016), no. 2, 1041–1054. MR 3466078, DOI 10.1016/j.jmaa.2016.02.039
Thomas Letendre, Expected volume and Euler characteristic of random submanifolds, J. Funct. Anal. 270 (2016), no. 8, 3047–3110. MR 3470435, DOI 10.1016/j.jfa.2016.01.007
T. Letendre, Variance of the volume of random real algebraic submanifolds. To appear in Trans. Amer. Math. Soc., arXiv:1608.05658.
T. Letendre and M. Puchol, Variance of the volume of random real algebraic submanifolds II, arXiv:1707.09771
J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275–280. MR 161339, DOI 10.1090/S0002-9939-1964-0161339-9
Ivan Nourdin and Giovanni Peccati, Normal approximations with Malliavin calculus, Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, Cambridge, 2012. From Stein’s method to universality. MR 2962301, DOI 10.1017/CBO9781139084659
Giovanni Peccati and Murad S. Taqqu, Wiener chaos: moments, cumulants and diagrams, Bocconi & Springer Series, vol. 1, Springer, Milan; Bocconi University Press, Milan, 2011. A survey with computer implementation; Supplementary material available online. MR 2791919, DOI 10.1007/978-88-470-1679-8
M. Shub and S. Smale, Complexity of Bezout’s theorem. II. Volumes and probabilities, Computational algebraic geometry (Nice, 1992) Progr. Math., vol. 109, Birkhäuser Boston, Boston, MA, 1993, pp. 267–285. MR 1230872, DOI 10.1007/978-1-4612-2752-6_{1}9
Mario Wschebor, On the Kostlan-Shub-Smale model for random polynomial systems. Variance of the number of roots, J. Complexity 21 (2005), no. 6, 773–789. MR 2182444, DOI 10.1016/j.jco.2005.05.005