Topological complexity of spiked random polynomials and finite-rank spherical integrals

We study the annealed complexity of Gaussian random homogeneous polynomials on the (N-1)-dimensional unit sphere in the presence of deterministic perturbations depending on fixed orthonormal vectors and external parameters. We derive variational formulas for the exponential asymptotics of the average number of critical points and local maxima. Our approach combines the Kac-Rice formula with determinant asymptotics for finite-rank perturbations of Gaussian Wigner matrices. In particular, the determinant analysis builds on recent results by [Guionnet, Husson 2022] on finite-rank spherical integrals, which we use to establish large deviation estimates for the largest eigenvalue of finite-rank Gaussian Wigner matrices. The resulting variational problems reveal a topological phase transition: above an explicit threshold in the external parameters, new zero-complexity regions emerge, corresponding to critical points with large correlation with the perturbation vectors. We also identify regions associated with critical points having large correlations with several vectors simultaneously; numerical evidence suggests that these critical points are more likely to be saddles than local maxima.