Distribution of critical points

MO question

Consider the inner product space \(P_k\) of polynomials of degree \(\le k\) on the unit sphere \(\mathbb S^{n-1}\subset \mathbb R^n\). Let \(p\sim N(0,\sigma^2 I)\) be a randomly chosen polynomial in \(P_k\), according to the standard Gaussian distribution. What is the average number of local minima of \(p\)?

Nicolaescu used the Kac-Rice formula to address the case where \(n\) is kept fixed and \(k\to \infty\). In fact, his result is more general: the manifold is fixed and the space of eigenfunctions of the Laplacian with eigenvalue \(\le L\) is considered.

However, I’m interested in the asymptotics when \(k\) is fixed (ex. take \(k=3\) or 4) and \(n\to \infty\), in order to help understand the tractability of random optimization problems as \(n\to \infty\). I’m also interested in any other asymptotics that are known, e.g. distribution of values at local minima, distribution of other critical points, etc.