Decomposition of the Kostlan–Shub–Smale model for random polynomials

Abstract

Let Pn be the space of homogeneous polynomials of degree n on R m+1 . We consider the asymptotic behavior of some coefficients relating to the decomposition of Pn into the sum of SO(m + 1)-irreducible components. Using the results, we prove that a random Kostlan-Shub-Smale polynomial u ∈ Pn can be approximated by polynomials of lower degree in the Sobolev spaces H k (S m ) on the unit sphere S m with small error and probability close to 1. For example, if ln > (m + 2k + 8ε)n ln n, then the inequality dist(u, P ln ) < An −ε u holds for any sufficiently large n with probability greater than 1 − Bn −2ε , where dist and are the distance and norm in H k (S m ), respectively, ε ∈ (0, 1), and A, B depend only on m and k. If ln > εn, then both the approximation error and the deviation of probability from 1 decay exponentially.

The Kostlan–Shub–Smale random polynomials in the case of growing number of variables

“…where ν j = c 2 j c 2 and c j = | ev Ej (p)|, c = | ev E (p)| (c j and c are independent of the choice of p ∈ M ). For the Kostlan-Shub-Smale model and the decomposition (1), the coefficients ν j,n were computed in [3]. As functions of t = j √ (m−1)n , j = 1, .…”