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

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