  We study the annealed complexity of a random Gaussian homogeneous polynomial
on the $N$-dimensional unit sphere in the presence of deterministic polynomials
that depend on fixed unit vectors and external parameters. In particular, we
establish variational formulas for the exponential asymptotics of the average
number of total critical points and of local maxima. This is obtained through
the Kac-Rice formula and the determinant asymptotics of a finite-rank
perturbation of a Gaussian Wigner matrix. More precisely, the determinant
analysis is based on recent advances on finite-rank spherical integrals by
[Guionnet, Husson 2022] to study the large deviations of multi-rank spiked
Gaussian Wigner matrices. The analysis of the variational problem identifies a
topological phase transition. There is an exact threshold for the external
parameters such that, once exceeded, the complexity function vanishes into new
regions in which the critical points are close to the given vectors.
Interestingly, these regions also include those where critical points are close
to multiple vectors.
