Geodesic
The Kostlan–Shub–Smale random polynomials in the case of growing number of variables
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 217-228 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $\mathcal{P}_n=\sum_{j}\mathcal{H}_{j}$ be the decomposition in $L^2(S^m)$ of the space of homogeneous polynomials of degree $n$ on $\mathbb{R}^{m+1}$ into the sum of irreducible components of the group $\mathrm{SO}(m+1)$. We consider the asymptotic behavior of the sequence $\nu_{n}(t)=\frac{\mathsf{E}(|\pi_{j}u|^{2})}{\mathsf{E}(|u|^{2})}$, where $t=\frac{j}{n}$, $\pi_{j}$ is the projection onto $\mathcal{H}_{j}$, and $\mathsf{E}$ stands for the expectation in the Kostlan-Shub–Smale model for random polynomials. Assuming $\frac{m}{n}\to a>0$ as $n\to\infty$, we prove that $\nu_{n}(t)$ is asymptotic to $\sqrt{\frac{4+a}{\pi n}}\,e^{-n(1+\frac{a}{4})(t-\sigma_{a})^{2}}$, where $\sigma_{a}=\frac12(\sqrt{a^{2}+4a}-a)$.
Keywords: random polynomials. 
@article{SEMR_2019_16_a128,
     author = {V. Gichev},
     title = {The {Kostlan{\textendash}Shub{\textendash}Smale} random polynomials in the case of growing number of variables},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {217--228},
     year = {2019},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2019_16_a128/}
}
TY  - JOUR
AU  - V. Gichev
TI  - The Kostlan–Shub–Smale random polynomials in the case of growing number of variables
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2019
SP  - 217
EP  - 228
VL  - 16
UR  - http://geodesic.mathdoc.fr/item/SEMR_2019_16_a128/
LA  - en
ID  - SEMR_2019_16_a128
ER  - 
%0 Journal Article
%A V. Gichev
%T The Kostlan–Shub–Smale random polynomials in the case of growing number of variables
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2019
%P 217-228
%V 16
%U http://geodesic.mathdoc.fr/item/SEMR_2019_16_a128/
%G en
%F SEMR_2019_16_a128
V. Gichev. The Kostlan–Shub–Smale random polynomials in the case of growing number of variables. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 217-228. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a128/

[1] Fyodorov, Y., Lerario, A., Lundberg E., “On the number of connected components of random algebraic hypersurfaces”, J. Geometry and Physics, 95 (2015), 1–20 | DOI | MR | Zbl

[2] Gichev V. M., “Metric properties in the mean of polynomials on compact isotropy irreducible homogeneous spaces”, Analysis and Math. Physics, 3:2 (2013), 119–144 | DOI | MR | Zbl

[3] V. Gichev, “Decomposition of the Kostlan-Shub-Smale model for random polynomials”, Contemporary Mathematics, 699, Amer. Math. Soc., Providence, RI, 2017, 103–120 | DOI | MR | Zbl

[4] Kostlan E., “On the distribution of roots of random polynomials”, The work of Smale in differential topology, From Topology to Computation: Proceedings of the Smalefest, Springer, 1993, 419–431 | DOI | MR | Zbl

[5] Kostlan E., “On the expected number of real roots of a system of random polynomial equations”, Foundations of computational mathematics, Proceedings of Smalefest 2000, World Scientific Publishing, 2002, 149–188 | DOI | MR | Zbl

[6] Shub M., Smale S., “Complexity of Bezout’s theorem II: volumes and probabilities”, Computational Algebraic Geometry, Progress in Mathematics, 109, eds. F. Eyssette, A. Galligo, Birkhäuser, 1993, 267–285 | MR | Zbl

[7] Stein E. M., Singular integrals and differentiability properties of functions, Princeton, 1970 | MR