Critical sets of random smooth functions on products of spheres

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 $V\subset C^\infty(M)$ 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 $\leq \nu$ is a random variable $Z_\nu$ with expectation $E(Z_\nu)\sim 2\sqrt{0.6}\,\nu$ and variance $var(Z_\nu)\sim c\nu$ as $\nu\to \infty$, $c\approx 0.35$.