Obtain sharp mode and critical-point bounds for isotropic Gaussian mixtures

Determine, for every pair \((n,d)\), the sharp upper bounds on the numbers of critical points, modes, and global modes of an \(n\)-component isotropic Gaussian mixture in dimension \(d\).

Background

The paper proves qualitative finiteness of the critical set for every finite isotropic Gaussian mixture, thereby making previously conditional quantitative mode-counting bounds applicable. However, the functional-transcendence proof is non-effective and does not itself give an upper bound on the number of critical points or modes.

Existing general bounds are described as far from sharp, and the paper therefore explicitly asks for the exact extremal quantities as functions of the number of components $n$ and ambient dimension $d$. The author also states that no existing conjecture predicts these sharp bounds.