Random systems of polynomial equations. The expected number of roots under smooth analysis

Armentano, Diego; Wschebor, Mario

Bernoulli, Tome 15 (2009) no. 1, p. 249-266 / Harvested from Project Euclid

Résumé

We consider random systems of equations over the reals, with m equations and m unknowns P_i(t)+X_i(t)=0, t∈ℝ^m, i=1, …, m, where the P_i’s are non-random polynomials having degrees d_i’s (the “signal”) and the X_i’s (the “noise”) are independent real-valued Gaussian centered random polynomial fields defined on ℝ^m, with a probability law satisfying some invariance properties. ¶ For each i, P_i and X_i have degree d_i. ¶ The problem is the behavior of the number of roots for large m. We prove that under specified conditions on the relation signal over noise, which imply that in a certain sense this relation is neither too large nor too small, it follows that the quotient between the expected value of the number of roots of the perturbed system and the expected value corresponding to the centered system (i.e., P_i identically zero for all i=1, …, m), tends to zero geometrically fast as m tends to infinity. In particular, this means that the behavior of this expected value is governed by the noise part.

Publié le : 2009-02-15
Classification: random polynomials, Rice formula, system of random equations

@article{1233669890,
     author = {Armentano, Diego and Wschebor, Mario},
     title = {Random systems of polynomial equations. The expected number of roots under smooth analysis},
     journal = {Bernoulli},
     volume = {15},
     number = {1},
     year = {2009},
     pages = { 249-266},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1233669890}
}

Armentano, Diego; Wschebor, Mario. Random systems of polynomial equations. The expected number of roots under smooth analysis. Bernoulli, Tome 15 (2009) no. 1, pp.  249-266. http://gdmltest.u-ga.fr/item/1233669890/