On the Kostlan-Shub-Smale model for random polynomial systems.: variance of the number of roots

  • Authors:

  • Mario Wschebor

  • Affiliations:

  • Centro de Matemática, Facultad de Ciencias, Universidad de la Republica, Calle Iguá, Montevideo, Uruguay

  • Venue:
  • Journal of Complexity
  • Year:
  • 2005

Quantified Score

Hi-index 0.00

Visualization

Abstract

We consider a random polynomial system with m equations and m real unknowns. Assume all equations have the same degree d and the law on the coefficients satisfies the Kostlan-Shub-Smale hypotheses. It is known that E(NX) = dm/2 where NX denotes the number of roots of the system. Under the condition that d does not grow very fast, we prove that lim supm→+∞ Var(NX/dm/2) ≤ 1. Moreover, if d ≥ 3 then Var(NX/dm/2) → 0 as m → +∞, which implies NX/dm/2 → 1 in probability.