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

Abstract

j Hj be the decomposition in L 2 (S m ) of the space of homogeneous polynomials of degree n on R m+1 into the sum of irreducible components of the group SO(m + 1). We consider the asymptotic behavior of the sequence νn(t) =n , πj is the projection onto Hj, and E stands for the expectation in the Kostlan-Shub-Smale model for random polynomials. Assuming m n → a > 0 as n → ∞, we prove that νn(t) is asymptotic to √ 4+a πn e −n(1+ a 4 )(t−σa) 2 , where σa = 1 2 ( √ a 2 + 4a − a). Keywords: random polynomials. Gichev, V.M., The Kostlan-Shub-Smale random polynomials in the case of growing number of variables. c ⃝ 2019 Gichev V.M. The work is supported by the program of fundamental researches of