Critical sets of random smooth functions on products of spheres

arXiv1008.5085MaRDI QIDQ6220445

Author name not available (Why is that?)

Publication date: 30 August 2010

Abstract: We prove a Chern-Lashof type formula computing the expected number of critical points of smooth function on a smooth manifold

M

randomly chosen from a finite dimensional subspace

V s u b s e t C^{i} n f t y (M)

equipped with a Gaussian probability measure. We then use this formula this formula to find the asymptotics of the expected number of critical points of a random linear combination of a large number eigenfunctions of the Laplacian on the round sphere, tori, or a products of two round spheres. In the case

M = S^{1}

we show that the number of critical points of a trigonometric polynomial of degree

l e q u

is a random variable

Z_{u}

with expectation

E (Z_{u}) s i m 2 s q r t 0.6, u

and variance

v a r (Z_{u}) s i m c u

as

u o i n f t y

,

c a p p r o x 0.35

.

Mathematics Subject Classification ID

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