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.
