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


Authors: D. Armentano, J-M. Azaïs, F. Dalmao and J. R. León
Journal: Proc. Amer. Math. Soc. 146 (2018), 5437-5449
MSC (2010): Primary 60F05, 30C15; Secondary 60G60, 65H10
DOI: https://doi.org/10.1090/proc/14215
Published electronically: September 17, 2018
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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.


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Additional Information

D. Armentano
Affiliation: CMAT, Universidad de la República, Montevideo, Uruguay
Email: diego@cmat.edu.uy

J-M. Azaïs
Affiliation: IMT, UMR CNRS 5219, Université de Toulouse, 31400 Toulouse, France
Email: jean-marc.azais@math.univ-toulouse.fr

F. Dalmao
Affiliation: DMEL, Universidad de la República, 50000 Salto, Uruguay
Email: fdalmao@unorte.edu.uy

J. R. León
Affiliation: IMERL, Universidad de la República, Montevideo, Uruguay — and — Escuela de Matemática, Facultad de Ciencias, Universidad Central de Venezuela, Caracas, Venezuela
Email: rlramos@fing.edu.uy

DOI: https://doi.org/10.1090/proc/14215
Received by editor(s): April 20, 2017
Received by editor(s) in revised form: April 10, 2018
Published electronically: September 17, 2018
Additional Notes: The first and third authors were partially supported by Agencia Nacional de Investigación e Innovación (ANII), Uruguay.
The first author was partially supported by CSIC group 618.
Communicated by: Zhen-Qing Chen
Article copyright: © Copyright 2018 American Mathematical Society

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